Vectors PDF

Introduction To Vectors Vector Products VECTORS Dr. Gabriel Obed Fosu February 28, 2022 Dr. Gabriel Obed Fosu 1/38  Introduction To Vectors Vector Products Outline 1 Introduction To Vectors...

Introduction To Vectors Vector Products VECTORS Dr. Gabriel Obed Fosu February 28, 2022 Dr. Gabriel Obed Fosu 1/38  Introduction To Vectors Vector Products Outline 1 Introduction To Vectors Introduction Vectors in Rn 2 Vector Products Dot Product Dr. Gabriel Obed Fosu 2/38  Introduction To Vectors Introduction Vector Products Vectors in Rn Outline of Presentation 1 Introduction To Vectors Introduction Vectors in Rn 2 Vector Products Dot Product Dr. Gabriel Obed Fosu 3/38  Introduction To Vectors Introduction Vector Products Vectors in Rn Introduction To Vectors Definition (Vector) A vector is an object that has both a magnitude and a direction. Dr. Gabriel Obed Fosu 4/38  Introduction To Vectors Introduction Vector Products Vectors in Rn Introduction To Vectors Definition (Vector) A vector is an object that has both a magnitude and a direction. 1 An example of a vector quantity is velocity. This is speed, in a particular direction. An example of velocity might be 60 mph due north. Dr. Gabriel Obed Fosu 4/38  Introduction To Vectors Introduction Vector Products Vectors in Rn Introduction To Vectors Definition (Vector) A vector is an object that has both a magnitude and a direction. 1 An example of a vector quantity is velocity. This is speed, in a particular direction. An example of velocity might be 60 mph due north. 2 A quantity with magnitude alone, but no direction, is not a vector. It is called a scalar instead. One example of a scalar is distance. Dr. Gabriel Obed Fosu 4/38  Introduction To Vectors Introduction Vector Products Vectors in Rn Geometric representation Definition (Geometric representation) −−→ A vector v is represented by a directed line segment denoted by AB. A −→ v = AB B 1 A is the initial point/origin/tail and B is the terminal point/endpoint/tip. 2 The length of the segment is the magnitude of v and is denoted by |v|. 3 A and B are any points in space. −−→ Vectors are represented with arrows on top (~v or OP ) or as boldface v. Dr. Gabriel Obed Fosu 5/38  Introduction To Vectors Introduction Vector Products Vectors in Rn Definition Two vectors are equal if they have the same magnitude and direction. Dr. Gabriel Obed Fosu 6/38  Introduction To Vectors Introduction Vector Products Vectors in Rn Definition Two vectors are equal if they have the same magnitude and direction. Example D B u u C u A −→ −−→ AB = u = DC Dr. Gabriel Obed Fosu 6/38  Introduction To Vectors Introduction Vector Products Vectors in Rn Addition of Vectors Theorem (Parallelogram law) Vector u + v is the diagonal of the parallelogram formed by u and v. Addition Laws B u A u+v D v C (a) Parallelogram law of vector addition: The tail of u and v coincide. Dr. Gabriel Obed Fosu 7/38  Introduction To Vectors Introduction Vector Products Vectors in Rn Addition of Vectors Theorem (Parallelogram law) Vector u + v is the diagonal of the parallelogram formed by u and v. Addition Laws B u B A u+v D u v A u+v v D C (b) Triangle law of vector addition: The tip of u coincides with the tail (a) Parallelogram law of vector of v. (Also called head to tail rule) addition: The tail of u and v coincide. Dr. Gabriel Obed Fosu 7/38  Introduction To Vectors Introduction Vector Products Vectors in Rn Addition If the two vectors do not have a common point, we can always coincide them by shifting one of the vectors. By observing that −−→ −−→ −−→ AB + BD = AD (1) Dr. Gabriel Obed Fosu 8/38  Introduction To Vectors Introduction Vector Products Vectors in Rn Addition If the two vectors do not have a common point, we can always coincide them by shifting one of the vectors. By observing that −−→ −−→ −−→ AB + BD = AD (1) Zero Vector −−→ −−→ −→ 1 AB + BA = AA. This is the zero vector. It has zero magnitude and is → − denoted by 0 or 0. Its origin is equal to its endpoint. −−→ −−→ 2 For any vector w, w + 0 = w [if we let w = M N and 0 = N N then −−→ −−→ −−→ w + 0 = M N + N N = M N = w.] Dr. Gabriel Obed Fosu 8/38  Introduction To Vectors Introduction Vector Products Vectors in Rn Addition If the two vectors do not have a common point, we can always coincide them by shifting one of the vectors. By observing that −−→ −−→ −−→ AB + BD = AD (1) Zero Vector −−→ −−→ −→ 1 AB + BA = AA. This is the zero vector. It has zero magnitude and is → − denoted by 0 or 0. Its origin is equal to its endpoint. −−→ −−→ 2 For any vector w, w + 0 = w [if we let w = M N and 0 = N N then −−→ −−→ −−→ w + 0 = M N + N N = M N = w.] Negative vector −−→ −−→ −−→ BA and AB have the same magnitude but opposite directions and satisfy AB + −−→ −→ ~ −−→ −−→ −−→ −−→ BA = AA = 0. BA is the negative of AB i.e. BA = −AB. Dr. Gabriel Obed Fosu 8/38  Introduction To Vectors Introduction Vector Products Vectors in Rn −−→ −−→ −−→ −−→ −−→ 1 + · · · + AB} = k AB In general, we have k AB. For instance |AB + AB{z k if k ∈ Z. The coefficient k is a scalar and the product between a scalar and a vector is called scalar multiplication. Dr. Gabriel Obed Fosu 9/38  Introduction To Vectors Introduction Vector Products Vectors in Rn −−→ −−→ −−→ −−→ −−→ 1 + · · · + AB} = k AB In general, we have k AB. For instance |AB + AB{z k if k ∈ Z. The coefficient k is a scalar and the product between a scalar and a vector is called scalar multiplication. 2 w and kw are said to be parallel; they have the same direction if k > 0 and opposite direction if k < 0. Dr. Gabriel Obed Fosu 9/38  Introduction To Vectors Introduction Vector Products Vectors in Rn Position Vectors Definition (Position Vectors) While vectors can exist anywhere in space, a point is always defined relative to the origin, O. Vectors defined from the origin are called Position Vectors −−→ −→ −−→ AB = AO + OB (2) = [−~a] + ~b (3) = ~b − ~a (4) −−→ −→ = OB − OA (5) Dr. Gabriel Obed Fosu 10/38 Introduction To Vectors Introduction Vector Products Vectors in Rn It is natural to represent position vectors using coordinates. For example, in A = −→ (3, 2) and we write the vector ~a = OA = [3, 2] using square brackets. Similarly, ~b = [−1, 3] and ~c = [2, −1] Dr. Gabriel Obed Fosu 11/38  Introduction To Vectors Introduction Vector Products Vectors in Rn It is natural to represent position vectors using coordinates. For example, in A = −→ (3, 2) and we write the vector ~a = OA = [3, 2] using square brackets. Similarly, ~b = [−1, 3] and ~c = [2, −1] 1 The individual coordinates [3 and 2 in the case of ~a] are called the components of the vector. Dr. Gabriel Obed Fosu 11/38  Introduction To Vectors Introduction Vector Products Vectors in Rn It is natural to represent position vectors using coordinates. For example, in A = −→ (3, 2) and we write the vector ~a = OA = [3, 2] using square brackets. Similarly, ~b = [−1, 3] and ~c = [2, −1] 1 The individual coordinates [3 and 2 in the case of ~a] are called the components of the vector. 2 A vector is sometimes said to be an ordered pair of real numbers. That is [3, 2] 6= [2, 3] (6) Dr. Gabriel Obed Fosu 11/38  Introduction To Vectors Introduction Vector Products Vectors in Rn It is natural to represent position vectors using coordinates. For example, in A = −→ (3, 2) and we write the vector ~a = OA = [3, 2] using square brackets. Similarly, ~b = [−1, 3] and ~c = [2, −1] 1 The individual coordinates [3 and 2 in the case of ~a] are called the components of the vector. 2 A vector is sometimes said to be an ordered pair of real numbers. That is [3, 2] 6= [2, 3] (6) 3 Vectorscan  be represented either as row vector [a, b, c] of a column a vector  b  c Dr. Gabriel Obed Fosu 11/38  Introduction To Vectors Introduction Vector Products Vectors in Rn Some 2D Properties Given u = [u1 , u2 ] and v = [v1 , v2 ] then 1 Addition u + v = [u1 + v1 , u2 + v2 ] (7) Dr. Gabriel Obed Fosu 12/38  Introduction To Vectors Introduction Vector Products Vectors in Rn Some 2D Properties Given u = [u1 , u2 ] and v = [v1 , v2 ] then 1 Addition u + v = [u1 + v1 , u2 + v2 ] (7) 2 Subtraction u − v = [u1 − v1 , u2 − v2 ] (8) Dr. Gabriel Obed Fosu 12/38  Introduction To Vectors Introduction Vector Products Vectors in Rn Some 2D Properties Given u = [u1 , u2 ] and v = [v1 , v2 ] then 1 Addition u + v = [u1 + v1 , u2 + v2 ] (7) 2 Subtraction u − v = [u1 − v1 , u2 − v2 ] (8) 3 Scalar multiplication ku = [ku1 , ku2 ] (9) Dr. Gabriel Obed Fosu 12/38  Introduction To Vectors Introduction Vector Products Vectors in Rn Some 2D Properties Given u = [u1 , u2 ] and v = [v1 , v2 ] then 1 Addition u + v = [u1 + v1 , u2 + v2 ] (7) 2 Subtraction u − v = [u1 − v1 , u2 − v2 ] (8) 3 Scalar multiplication ku = [ku1 , ku2 ] (9) 4 Magnitude / Length / Norm q |u| = ||u|| = u21 + u22 (10) Dr. Gabriel Obed Fosu 12/38  Introduction To Vectors Introduction Vector Products Vectors in Rn Standard Basis Vectors Let ~i = [1, 0] and ~j = [0, 1] , the i, j are called standard basis vectors in R2. Each vector have length 1 and point in the directions of the positive x, and y−axes respectively. Dr. Gabriel Obed Fosu 13/38  Introduction To Vectors Introduction Vector Products Vectors in Rn Standard Basis Vectors Let ~i = [1, 0] and ~j = [0, 1] , the i, j are called standard basis vectors in R2. Each vector have length 1 and point in the directions of the positive x, and y−axes respectively. Similarly, in the three-dimensional plane, the vectors ~i = [1, 0, 0], ~j = [0, 1, 0] and ~k = [0, 0, 1] are also called the standard basis vectors. Again they have length 1 and point in the directions of the positive x, y, and z−axes. Dr. Gabriel Obed Fosu 13/38  Introduction To Vectors Introduction Vector Products Vectors in Rn Standard Basis Vectors Dr. Gabriel Obed Fosu 14/38  Introduction To Vectors Introduction Vector Products Vectors in Rn Unit Vector Definition 1 A vector of magnitude 1 is called a unit vector. In the Cartesian coordinate system, i and j are reserved for the unit vector along the positive x-axis and y- axis respectively. 2 The Cartesian coordinate system is therefore defined by three reference points (O, I, J) such that the origin −→ O has coordinates (0, 0), i = OI and −→ −−→ j = OJ, and any vector w = OM can be expressed as w = w1 i + w2 j. w1 and w2 are the coordinates of w. Dr. Gabriel Obed Fosu 15/38  Introduction To Vectors Introduction Vector Products Vectors in Rn Example −−→ Given the points E = (2, 7) and F = (3, −1), then the vector EF is given as −−→ −−→ −−→ EF = OF − OE (11) = [(3, −1) − (2, 7)] (12) = [1, −8] (13) Dr. Gabriel Obed Fosu 16/38  Introduction To Vectors Introduction Vector Products Vectors in Rn Example Calculate the magnitude of vector a = λ1 a1 − λ2 a2 − λ3 a3 in case a1 = [2, −3], a2 = [−3, 0], a3 = [−1, −1] and λ1 = 2, λ2 = −1, λ3 = 3. Dr. Gabriel Obed Fosu 17/38  Introduction To Vectors Introduction Vector Products Vectors in Rn Example Calculate the magnitude of vector a = λ1 a1 − λ2 a2 − λ3 a3 in case a1 = [2, −3], a2 = [−3, 0], a3 = [−1, −1] and λ1 = 2, λ2 = −1, λ3 = 3. a = λ1 a1 − λ2 a2 − λ3 a3 (14) = 2[2, −3] − [−1][−3, 0] − 3[−1, −1] (15) = [4, −6] + [−3, 0] + [3, 3] (16) = [4, −3] (17) Thus Dr. Gabriel Obed Fosu 17/38  Introduction To Vectors Introduction Vector Products Vectors in Rn Example Calculate the magnitude of vector a = λ1 a1 − λ2 a2 − λ3 a3 in case a1 = [2, −3], a2 = [−3, 0], a3 = [−1, −1] and λ1 = 2, λ2 = −1, λ3 = 3. a = λ1 a1 − λ2 a2 − λ3 a3 (14) = 2[2, −3] − [−1][−3, 0] − 3[−1, −1] (15) = [4, −6] + [−3, 0] + [3, 3] (16) = [4, −3] (17) Thus p √ |a| = 43 + [−3]2 = 25 = 5 (18) Dr. Gabriel Obed Fosu 17/38  Introduction To Vectors Introduction Vector Products Vectors in Rn Example A car is pulled by four men. The components of the four forces are F1 = [20, 25], F2 = [15, 5], F3 = [25, −5] and F4 = [30, −15]. Find the resultant force R. Dr. Gabriel Obed Fosu 18/38  Introduction To Vectors Introduction Vector Products Vectors in Rn Example A car is pulled by four men. The components of the four forces are F1 = [20, 25], F2 = [15, 5], F3 = [25, −5] and F4 = [30, −15]. Find the resultant force R. The resultant force F = F1 + F2 + F3 + F4 (19) = [20, 25] + [15, 5] + [25, −5] + [30, −15] (20) = [90, 10] (21) Dr. Gabriel Obed Fosu 18/38  Introduction To Vectors Introduction Vector Products Vectors in Rn Definition 1 If v , v ,... , v 1 2 n are n real numbers: 2 v = [v1 , v2 ] is a two dimension vector. It belongs to R2. 3 v = [v1 , v2 , v3 ] is a three dimension vector. It belongs to R3. 4 v = [v1 , v2 ,... , vn ] is an n dimension vector. It belongs to Rn. We may use e1 , e2 ,... en to denote the unit vectors of the coordinate system. Dr. Gabriel Obed Fosu 19/38  Introduction To Vectors Introduction Vector Products Vectors in Rn Definition 1 If v , v ,... , v 1 2 n are n real numbers: 2 v = [v1 , v2 ] is a two dimension vector. It belongs to R2. 3 v = [v1 , v2 , v3 ] is a three dimension vector. It belongs to R3. 4 v = [v1 , v2 ,... , vn ] is an n dimension vector. It belongs to Rn. We may use e1 , e2 ,... en to denote the unit vectors of the coordinate system. 1 i = [1, 0, 0], j = [0, 1, 0] and k = [0, 0, 1] are three dimension vectors; i, j, k ∈ R3.u = [3, −1, 5] = 3i − j + 5k is a vector in R3. Dr. Gabriel Obed Fosu 19/38  Introduction To Vectors Introduction Vector Products Vectors in Rn Definition 1 If v , v ,... , v 1 2 n are n real numbers: 2 v = [v1 , v2 ] is a two dimension vector. It belongs to R2. 3 v = [v1 , v2 , v3 ] is a three dimension vector. It belongs to R3. 4 v = [v1 , v2 ,... , vn ] is an n dimension vector. It belongs to Rn. We may use e1 , e2 ,... en to denote the unit vectors of the coordinate system. 1 i = [1, 0, 0], j = [0, 1, 0] and k = [0, 0, 1] are three dimension vectors; i, j, k ∈ R3.u = [3, −1, 5] = 3i − j + 5k is a vector in R3. 2 v = [3, −1, 5, 0, 4] = 3e1 − e2 + 5e3 + 4e5 is a five dimension vector; v ∈ R5. Dr. Gabriel Obed Fosu 19/38  Introduction To Vectors Introduction Vector Products Vectors in Rn Definition 1 If v , v ,... , v 1 2 n are n real numbers: 2 v = [v1 , v2 ] is a two dimension vector. It belongs to R2. 3 v = [v1 , v2 , v3 ] is a three dimension vector. It belongs to R3. 4 v = [v1 , v2 ,... , vn ] is an n dimension vector. It belongs to Rn. We may use e1 , e2 ,... en to denote the unit vectors of the coordinate system. 1 i = [1, 0, 0], j = [0, 1, 0] and k = [0, 0, 1] are three dimension vectors; i, j, k ∈ R3.u = [3, −1, 5] = 3i − j + 5k is a vector in R3. 2 v = [3, −1, 5, 0, 4] = 3e1 − e2 + 5e3 + 4e5 is a five dimension vector; v ∈ R5. 3 The coordinate system is defined by e1 = [1, 0, 0, 0, 0], e2 = [0, 1, 0, 0, 0], e3 = [0, 0, 1, 0, 0], e4 = [0, 0, 0, 1, 0], and e5 = [0, 0, 0, 0, 1]. Dr. Gabriel Obed Fosu 19/38  Introduction To Vectors Introduction Vector Products Vectors in Rn u = [u1 , u2 ,... , un ], and v = [v1 , v2 ,... , vn ] then 1 Addition u + v = [u1 + v1 , u2 + v2 ,... , un + vn ] (22) Dr. Gabriel Obed Fosu 20/38  Introduction To Vectors Introduction Vector Products Vectors in Rn u = [u1 , u2 ,... , un ], and v = [v1 , v2 ,... , vn ] then 1 Addition u + v = [u1 + v1 , u2 + v2 ,... , un + vn ] (22) 2 Subtraction u − v = [u1 − v1 , u2 − v2 ,... , un − vn ] (23) Dr. Gabriel Obed Fosu 20/38  Introduction To Vectors Introduction Vector Products Vectors in Rn u = [u1 , u2 ,... , un ], and v = [v1 , v2 ,... , vn ] then 1 Addition u + v = [u1 + v1 , u2 + v2 ,... , un + vn ] (22) 2 Subtraction u − v = [u1 − v1 , u2 − v2 ,... , un − vn ] (23) 3 Scalar multiplication ku = [ku1 , ku2 ,... , kun ] (24) Dr. Gabriel Obed Fosu 20/38  Introduction To Vectors Introduction Vector Products Vectors in Rn u = [u1 , u2 ,... , un ], and v = [v1 , v2 ,... , vn ] then 1 Addition u + v = [u1 + v1 , u2 + v2 ,... , un + vn ] (22) 2 Subtraction u − v = [u1 − v1 , u2 − v2 ,... , un − vn ] (23) 3 Scalar multiplication ku = [ku1 , ku2 ,... , kun ] (24) 4 Magnitude q |u| = u21 + u22 +... + u2n (25) Dr. Gabriel Obed Fosu 20/38  Introduction To Vectors Introduction Vector Products Vectors in Rn u = [u1 , u2 ,... , un ], and v = [v1 , v2 ,... , vn ] then 1 Addition u + v = [u1 + v1 , u2 + v2 ,... , un + vn ] (22) 2 Subtraction u − v = [u1 − v1 , u2 − v2 ,... , un − vn ] (23) 3 Scalar multiplication ku = [ku1 , ku2 ,... , kun ] (24) 4 Magnitude q |u| = u21 + u22 +... + u2n (25) 5 Unit vector in the direction of u   u u1 u2 un eu = = , ,..., (26) |u| |u| |u| |u| Dr. Gabriel Obed Fosu 20/38  Introduction To Vectors Introduction Vector Products Vectors in Rn Example If a = [4, 0, 3] and b = [−2, 1, 5], find |a − b| and the unit vector e in the direction of a. Dr. Gabriel Obed Fosu 21/38  Introduction To Vectors Introduction Vector Products Vectors in Rn Example If a = [4, 0, 3] and b = [−2, 1, 5], find |a − b| and the unit vector e in the direction of a. ~a − ~b = [4, 0, 3] − [−2, 1, 5] (27) = [6, −1, −2] (28) and √ |~a − ~b| = p 62 + [−1]2 + [−2]2 = 41 (29) Dr. Gabriel Obed Fosu 21/38  Introduction To Vectors Introduction Vector Products Vectors in Rn Example If a = [4, 0, 3] and b = [−2, 1, 5], find |a − b| and the unit vector e in the direction of a. ~a − ~b = [4, 0, 3] − [−2, 1, 5] (27) = [6, −1, −2] (28) and √ |~a − ~b| = p 62 + [−1]2 + [−2]2 = 41 (29) The unit in the direction of a is 1 e= √ [4, 0, 3] (30) 42 + 02 + 32 1 = [4, 0, 3] (31) 5  4 3 = , 0, (32) 5 5 Dr. Gabriel Obed Fosu 21/38  Introduction To Vectors Introduction Vector Products Vectors in Rn Properties of vectors For any vectors u, v, w ∈ Rn and any scalars k, k 0 ∈ R, 1 Closure u + v ∈ Rn Dr. Gabriel Obed Fosu 22/38  Introduction To Vectors Introduction Vector Products Vectors in Rn Properties of vectors For any vectors u, v, w ∈ Rn and any scalars k, k 0 ∈ R, 1 Closure u + v ∈ Rn 2 Additive inverse u + [−u] = 0 Dr. Gabriel Obed Fosu 22/38  Introduction To Vectors Introduction Vector Products Vectors in Rn Properties of vectors For any vectors u, v, w ∈ Rn and any scalars k, k 0 ∈ R, 1 Closure u + v ∈ Rn 2 Additive inverse u + [−u] = 0 3 Commutative u + v = v + u Dr. Gabriel Obed Fosu 22/38  Introduction To Vectors Introduction Vector Products Vectors in Rn Properties of vectors For any vectors u, v, w ∈ Rn and any scalars k, k 0 ∈ R, 1 Closure u + v ∈ Rn 2 Additive inverse u + [−u] = 0 3 Commutative u + v = v + u 4 Additive identity u + 0 = u Dr. Gabriel Obed Fosu 22/38  Introduction To Vectors Introduction Vector Products Vectors in Rn Properties of vectors For any vectors u, v, w ∈ Rn and any scalars k, k 0 ∈ R, 1 Closure u + v ∈ Rn 2 Additive inverse u + [−u] = 0 3 Commutative u + v = v + u 4 Additive identity u + 0 = u 5 Multiplicative identity 1u = u Dr. Gabriel Obed Fosu 22/38  Introduction To Vectors Introduction Vector Products Vectors in Rn Properties of vectors For any vectors u, v, w ∈ Rn and any scalars k, k 0 ∈ R, 1 Closure u + v ∈ Rn 2 Additive inverse u + [−u] = 0 3 Commutative u + v = v + u 4 Additive identity u + 0 = u 5 Multiplicative identity 1u = u 6 Associative [u + v] + w = u + [v + w] Dr. Gabriel Obed Fosu 22/38  Introduction To Vectors Introduction Vector Products Vectors in Rn Properties of vectors For any vectors u, v, w ∈ Rn and any scalars k, k 0 ∈ R, 1 Closure u + v ∈ Rn 2 Additive inverse u + [−u] = 0 3 Commutative u + v = v + u 4 Additive identity u + 0 = u 5 Multiplicative identity 1u = u 6 Associative [u + v] + w = u + [v + w] 7 Scalar Multiplication k[u + v] = ku + kv Dr. Gabriel Obed Fosu 22/38  Introduction To Vectors Introduction Vector Products Vectors in Rn Properties of vectors For any vectors u, v, w ∈ Rn and any scalars k, k 0 ∈ R, 1 Closure u + v ∈ Rn 2 Additive inverse u + [−u] = 0 3 Commutative u + v = v + u 4 Additive identity u + 0 = u 5 Multiplicative identity 1u = u 6 Associative [u + v] + w = u + [v + w] 7 Scalar Multiplication k[u + v] = ku + kv 8 Scalar Multiplication [k + k 0 ]u = ku + k 0 u Dr. Gabriel Obed Fosu 22/38  Introduction To Vectors Introduction Vector Products Vectors in Rn Definition (Linear Combination) A vector v is a linear combination of vectors v1 , v2 , · · · , vn if there are scalars k1 , k2 , · · · , kn such that v = k1 v1 + k2 v2 + · · · + kn vn (33) The scalars k1 , k2 , · · · , kn are called the coefficients of the linear combination. Dr. Gabriel Obed Fosu 23/38  Introduction To Vectors Introduction Vector Products Vectors in Rn Definition (Linear Combination) A vector v is a linear combination of vectors v1 , v2 , · · · , vn if there are scalars k1 , k2 , · · · , kn such that v = k1 v1 + k2 v2 + · · · + kn vn (33) The scalars k1 , k2 , · · · , kn are called the coefficients of the linear combination. Example         2 1 2 5 The vectors −2 is a linear combination of 0 , −3 and −4Since        1 −1 1 0         1 2 5 2 3  0  + 2 −3 − −4 = −2 (34) −1 1 0 1 Dr. Gabriel Obed Fosu 23/38  Introduction To Vectors Introduction Vector Products Vectors in Rn Exercise 1 Let M = (m , m ) and N = (n , n ) be two points in the Cartesian 1 2 1 2 −−→ −−→ −−→ coordinate system (O, I, J). Express u = M N , v = M O−2ON , w = −−→ −−→ −−→ −→ −→ −→ −→ M N + 2N M + ON − OI + 3IJ in terms of OI and OJ. 2 Find a unit vector that has the same direction as a. u = 8i − j + k b. v = [−2, 1, 2]. √ √ 3 Show that u = 2e1 −4e2 +e3 − 2e4 and v = −4e1 +8e2 −2e3 + 8e4 are parallel vectors. Dr. Gabriel Obed Fosu 24/38  Introduction To Vectors Dot Product Vector Products Outline of Presentation 1 Introduction To Vectors Introduction Vectors in Rn 2 Vector Products Dot Product Dr. Gabriel Obed Fosu 25/38  Introduction To Vectors Dot Product Vector Products Dot Product For u and v in Rn , 1 If u = [u1 , u2 ,... , un ], and v = [v1 , v2 ,... , vn ] then hu, vi = u · v = u1 v1 + u2 v2 +... + un vn (35) is the dot product of the two vectors. 2 The dot product is also called inner or scalar product. Dr. Gabriel Obed Fosu 26/38  Introduction To Vectors Dot Product Vector Products Dot Product For u and v in Rn , 1 If u = [u1 , u2 ,... , un ], and v = [v1 , v2 ,... , vn ] then hu, vi = u · v = u1 v1 + u2 v2 +... + un vn (35) is the dot product of the two vectors. 2 The dot product is also called inner or scalar product. Example [1, 2, 3]·[−1, 0, 1] = 1(−1) + 2(0) + 3(1) = 2. Dr. Gabriel Obed Fosu 26/38  Introduction To Vectors Dot Product Vector Products Dot Product For u and v in Rn , 1 If u = [u1 , u2 ,... , un ], and v = [v1 , v2 ,... , vn ] then hu, vi = u · v = u1 v1 + u2 v2 +... + un vn (35) is the dot product of the two vectors. 2 The dot product is also called inner or scalar product. Example [1, 2, 3]·[−1, 0, 1] = 1(−1) + 2(0) + 3(1) = 2. Example [i + 2j − 3k] · [2j − k] = 0 + 4 + 3 = 7. Dr. Gabriel Obed Fosu 26/38  Introduction To Vectors Dot Product Vector Products Example In R3 , i = [1, 0, 0], j = [0, 1, 0] and k = [0, 0, 1] so that · i j k i 1 0 0 j 0 1 0 k 0 0 1 Dr. Gabriel Obed Fosu 27/38  Introduction To Vectors Dot Product Vector Products Properties For any vectors u, v, w ∈ Rn and any scalar k ∈ R : 1 u·v ∈R Dr. Gabriel Obed Fosu 28/38  Introduction To Vectors Dot Product Vector Products Properties For any vectors u, v, w ∈ Rn and any scalar k ∈ R : 1 u·v ∈R 2 0 · u = 0. Dr. Gabriel Obed Fosu 28/38  Introduction To Vectors Dot Product Vector Products Properties For any vectors u, v, w ∈ Rn and any scalar k ∈ R : 1 u·v ∈R 2 0 · u = 0. 3 u·v =v·u Dr. Gabriel Obed Fosu 28/38  Introduction To Vectors Dot Product Vector Products Properties For any vectors u, v, w ∈ Rn and any scalar k ∈ R : 1 u·v ∈R 2 0 · u = 0. 3 u·v =v·u 4 u · u = |u|2 ≥ 0 and u · u = 0 iff u = 0. Dr. Gabriel Obed Fosu 28/38  Introduction To Vectors Dot Product Vector Products Properties For any vectors u, v, w ∈ Rn and any scalar k ∈ R : 1 u·v ∈R 2 0 · u = 0. 3 u·v =v·u 4 u · u = |u|2 ≥ 0 and u · u = 0 iff u = 0. 5 [ku]·v = k[u · v] Dr. Gabriel Obed Fosu 28/38  Introduction To Vectors Dot Product Vector Products Properties For any vectors u, v, w ∈ Rn and any scalar k ∈ R : 1 u·v ∈R 2 0 · u = 0. 3 u·v =v·u 4 u · u = |u|2 ≥ 0 and u · u = 0 iff u = 0. 5 [ku]·v = k[u · v] 6 u·[v + w] = u · v + u · w. Dr. Gabriel Obed Fosu 28/38  Introduction To Vectors Dot Product Vector Products Angles The dot product can also be used to calculate the angle between a pair of vectors. In R2 or R3 , the angle between the nonzero vectors u and v will refer to the angle θ determined by these vectors that satisfies 0 ≤ θ ≤ 180. Dr. Gabriel Obed Fosu 29/38 Introduction To Vectors Dot Product Vector Products Definition For nonzero vectors u and v in Rn u·v cos θ = (36) ||u|| ||v|| This implies that u · v = ||u|| ||v|| cos θ (37) 0 ≤ θ ≤ π is the internal angle. Dr. Gabriel Obed Fosu 30/38  Introduction To Vectors Dot Product Vector Products Definition For nonzero vectors u and v in Rn u·v cos θ = (36) ||u|| ||v|| This implies that u · v = ||u|| ||v|| cos θ (37) 0 ≤ θ ≤ π is the internal angle. Orthogonal vectors Two nonzero vectors u and v are said to be orthogonal or perpendicular if u·v =0 (38) Dr. Gabriel Obed Fosu 30/38  Introduction To Vectors Dot Product Vector Products Example Let u = i − 2j + 3k, v = i − k and w = 2i + 7j + 4k be three vectors of R3. Show that u and w are orthogonal but u and v are not. Find the angle θ between the direction of u and v. Dr. Gabriel Obed Fosu 31/38  Introduction To Vectors Dot Product Vector Products Example Let u = i − 2j + 3k, v = i − k and w = 2i + 7j + 4k be three vectors of R3. Show that u and w are orthogonal but u and v are not. Find the angle θ between the direction of u and v. u · v = [1, −2, 3] · [1, 0, −1] (39) = 1 + 0 − 3 = −2 Not orthogonal (40) Dr. Gabriel Obed Fosu 31/38  Introduction To Vectors Dot Product Vector Products Example Let u = i − 2j + 3k, v = i − k and w = 2i + 7j + 4k be three vectors of R3. Show that u and w are orthogonal but u and v are not. Find the angle θ between the direction of u and v. u · v = [1, −2, 3] · [1, 0, −1] (39) = 1 + 0 − 3 = −2 Not orthogonal (40) u · w = [1, −2, 3] · [2, 7, 4] (41) = 2 − 14 + 12 = 0 orthogonal (42) Dr. Gabriel Obed Fosu 31/38  Introduction To Vectors Dot Product Vector Products Example Let u = i − 2j + 3k, v = i − k and w = 2i + 7j + 4k be three vectors of R3. Show that u and w are orthogonal but u and v are not. Find the angle θ between the direction of u and v. u · v = [1, −2, 3] · [1, 0, −1] (39) = 1 + 0 − 3 = −2 Not orthogonal (40) u · w = [1, −2, 3] · [2, 7, 4] (41) = 2 − 14 + 12 = 0 orthogonal (42) u·v −2 −1 cos θ = =√ √ =√ (43) ||u|| ||v|| 14 × 2 7 θ = 112.2078 (44) Dr. Gabriel Obed Fosu 31/38  Introduction To Vectors Dot Product Vector Products Example Find k so that u = [1, k, −3, 1] and v = [1, k, k, 1] are orthogonal. Dr. Gabriel Obed Fosu 32/38  Introduction To Vectors Dot Product Vector Products Example Find k so that u = [1, k, −3, 1] and v = [1, k, k, 1] are orthogonal. For orthogonal u·v =0 (45) [1, k, −3, 1] · [1, k, k, 1] = 0 (46) 2 1 + k − 3k + 1 = 0 (47) 2 k − 3k + 2 = 0 (48) [k − 2][k − 1] = 0 (49) Hence k = 2 or k = 1 Dr. Gabriel Obed Fosu 32/38 Introduction To Vectors Dot Product Vector Products Theorem For any two vectors a and b 1 Schwartz’s inequality: |a · b| ≤ |a| |b| (50) 2 Triangle inequality: |a + b| ≤ |a| + |b| (51) Dr. Gabriel Obed Fosu 33/38  Introduction To Vectors Dot Product Vector Products Theorem For any two vectors a and b 1 Schwartz’s inequality: |a · b| ≤ |a| |b| (50) 2 Triangle inequality: |a + b| ≤ |a| + |b| (51) Definition The distance between a and b is d(a, b) = |a − b| (52) Dr. Gabriel Obed Fosu 33/38  Introduction To Vectors Dot Product Vector Products Theorem (Pythagoras) For all vectors u and v in Rn , |u + v|2 = |u|2 + |v|2 (53) if and only if u and v are orthogonal. Note |u + v|2 = |u|2 + |v|2 + 2u · v (54) For orthogonal u · v = 0 Dr. Gabriel Obed Fosu 34/38  Introduction To Vectors Dot Product Vector Products Projection of v onto u Consider two nonzero vectors uand v. Let p be the vector obtained by dropping a perpendicular from the head of vonto u and let θ be the angle between u and v Scaler Projection The scalar projection of a vector v onto a nonzero vector u is defined and denoted by the scalar u·v compu v = (55) |u| Dr. Gabriel Obed Fosu 35/38  Introduction To Vectors Dot Product Vector Products Projection of v onto u Consider two nonzero vectors uand v. Let p be the vector obtained by dropping a perpendicular from the head of vonto u and let θ be the angle between u and v Scaler Projection The scalar projection of a vector v onto a nonzero vector u is defined and denoted by the scalar u·v compu v = (55) |u| Vector Projection The [vector] projection of v onto u is u·v u u proju v = = compu v (56) |u| |u| |u| Dr. Gabriel Obed Fosu 35/38  Introduction To Vectors Dot Product Vector Products Example Find the projection of v onto u if u = [2, 1] and v = [−1, 3] Dr. Gabriel Obed Fosu 36/38  Introduction To Vectors Dot Product Vector Products Example Find the projection of v onto u if u = [2, 1] and v = [−1, 3] u · v = −2 + 3 = 1 (57) √ √ ||u|| · ||u|| = 5( 5) = 5 (58) Dr. Gabriel Obed Fosu 36/38  Introduction To Vectors Dot Product Vector Products Example Find the projection of v onto u if u = [2, 1] and v = [−1, 3] u · v = −2 + 3 = 1 (57) √ √ ||u|| · ||u|| = 5( 5) = 5 (58) u·v u 1 proju v = = u (59) |u| |u| 5 1 = [2, 1] (60) 5   2 1 = , (61) 5 5 Dr. Gabriel Obed Fosu 36/38  Introduction To Vectors Dot Product Vector Products Exercise Find the vector projection of v onto u 1 u = [−1, 1], v = [−2, 4] 2 u = [3/5, −4/5], v = [1, 2] 3 u = [1/2, −1/4, / − 1/2], v = [2, 2, / − 2] Dr. Gabriel Obed Fosu 37/38 Introduction To Vectors Vector Products END OF LECTURE THANK YOU Dr. Gabriel Obed Fosu 38/38