Product of Vectors Class XI PDF

# Vectors ## The magnitude of force The magnitude of force is the product of mass (m) and magnitude of acceleration a, ie., $F = ma$ or $F = m/a$ . The SI unit of force is $kg - m/s^2$, which is the product of unit of mass kg and unit of acceleration $m/s^2$. ## Product of Two Vectors The vect...

# Vectors ## The magnitude of force The magnitude of force is the product of mass (m) and magnitude of acceleration a, ie., $F = ma$ or $F = m/a$ . The SI unit of force is $kg - m/s^2$, which is the product of unit of mass kg and unit of acceleration $m/s^2$. ## Product of Two Vectors The vector quantities have magnitude as well as direction. Due to the directions of vectors, their product cannot be obtained by simple algebraic methods. The product of two vectors may be a scalar or a vector. For example, the product of force and position vectors yields torque which is a vector quantity, while the product of force and position vector yields work which is a scalar quantity. There are two types of products of two vectors: 1. **Scalar product:** If the product of two vectors yields a scalar quantity, the product is called scalar product. It is represented by a dot between the vectors like $A \cdot B$ (read as A dot B). Therefore, scalar product is also called dot product. 2. **Vector product:** If the product of two vectors yields a vector quantity, the product is called vector product. It is represented by a cross between the vectors like $A \times B$ (read as A cross B). Therefore, the vector product is also called cross product. ## Scalar Product of Two Vectors The scalar product of two vectors is defined as a scalar quantity whose value is the product of magnitudes of given vectors and cosine of the angle between them. If $A$ and $B$ are two vectors and $\theta$ is the angle between them, then by definition the scalar product of $A$ and $B$ is given by: $A \cdot B = AB cos \theta$ Here, A and B are magnitudes of vectors A and B respectively. Equation (1) may also be expressed as, $A \cdot B = A (B cos \theta)$ where $B cos \theta = ON$  is the projection of vector B along the direction of vector A. Accordingly, the scalar product of two vectors may also be defined as: The scalar product of two vectors is a scalar quantity whose value is equal to the product of magnitude of one vector and the projection of other vector in the direction of the first vector. ### Characteristics of Scalar Products 1. **Commutative Law:** If angle between two vectors A and B is $\theta$, then: $A \cdot B = AB cos \theta$ $B \cdot A = BA cos (360° - \theta)$ $B \cdot A = BA cos (- \theta)$ $B \cdot A = - AB cos \theta$ $A \cdot B = - B \cdot A$ This means that the vector product of two vectors does not obey commutative law. 2. **Distributive Law:** The scalar product of vectors obeys the distributive law, i.e., $A (B + C) = AB + AC$ 3. **Scalar Product of Two Mutually Perpendicular Vectors:** If two vectors A and B are mutually perpendicular, then $0 = 90°$, so the scalar product, $A \cdot B = AB cos 90° = 0$ Thus, the scalar product of two mutually perpendicular vectors is always zero. 4. **Scalar Product of Two Parallel Vectors:** If A and B are two parallel vectors, then angle between them $\theta = 0°$ $A \cdot B = AB cos 0° = AB$ Thus, the scalar product of two parallel vectors is equal to the product of their magnitudes. 5. **Scalar Self Product of a Vector:** If a vector A multiplied scalarly with itself, the result is the scalar self product of vector A. $A \cdot A = AA cos 0° = A^2$. Thus, the scalar self product of a vector is equal to the square of the magnitude of that vector. By this fact we may find the magnitude of resultant of two vectors analytically. If $R$ is the vector sum of two vectors A and B, then $R= A + B$. Taking scalar self product of both sides, we get $R \cdot R = (A + B) * (A + B) = A \cdot A + A \cdot B + B \cdot A + B \cdot B$ $A \cdot B = B \cdot A$ $R \cdot R = A \cdot A + A \cdot B + B \cdot A + B \cdot B$ If A and B are magnitudes of A, B and R, we have $R \cdot R = R^2$, $A \cdot A = A^2$, $B \cdot B = B^2$ and $A \cdot B = AB cos \theta$ $R^2 = A^2 + B^2 + 2 AB cos \theta$ $R = \sqrt{A^2 + B^2 + 2 AB cos \theta}$ ### Important Remarks 1. As unit vectors i, j and k along X, Y and Z axes are mutually perpendicular, therefore, $i \cdot j = j \cdot k = k \cdot i = 0$. 2. If the force F acting on a particle is perpendicular to its displacement s, then the work done by the force is always zero, i.e. $W = F \cdot s = Fs cos 90° = 0$ 3. If $A \cdot B = 0$, then it's not necessarily that $A$ and $B$ be mutually perpendicular, since either of $A$ or $B$ could be a null vector. ## Vector Product of Two Vectors The vector product of two vectors is a vector quantity whose magnitude is the product of magnitudes of the given vectors and the sine of the angle between their directions and whose direction is perpendicular to the plane of the vectors, given by right-handed screw rule. If $A$ and $B$ are two vectors and $ \theta$ is the angle between them, then their vector product of $A$ and $B$ is: $A \times B = AB sin \theta n$ where $n$ is a unit vector along $(A × B)$, i.e., perpendicular to the plane of A and B, given by the right-handed screw rule. The rule states that if A is turned towards B through a very small angle, then the direction of advance of a right-handed screw placed at O, gives the direction of $A \times B$ (or unit vector $n$). ### Examples of Vector Product of Two Vectors 1. **Torque:** The torque acting at a particle about an arbitary point is equal to the vector product of position vector (r) of that particle relative to that point and the force F acting at that point, i.e. $T = r \times F$ 2. **Angular Momentum:** The angular momentum of a particle relative to an arbitrary point is equal to the vector product of position vector of that particle relative to that point and the linear momentum (p) of the particle, i.e. $L = r \times p$ 3. **Linear Velocity:** Linear velocity of a point having position vector r on a rigid body rotating with angular velocity vector w is $v = w \times r$ ### Characteristics of Vector Products 1. **Commutative Law:** If A and B are two vectors and $\theta$, the angle between them, then $A \times B = AB sin \theta n$ $B \times A = BA sin (360° – \theta)$ $B \times A = BA sin (- \theta) n$ $B \times A = - AB sin \theta n$ $A \times B = - B \times A$ Thus, the vector product of two vectors does not obey commutative law. 2. **Vector Product of Parallel Vectors:** If A and B are parallel vectors, then angle between them $\theta = 0°$ $A \times B = AB sin 0º n = 0$ That is the vector product of parallel vectors is always a null vector 3. **Self vector product of a vector:** If vector product of vector A is taken with itself, then $\theta = 0°$ $A \times A = AA sin 0° = 0$ That is the self vector product of a vector is always a null vector. In particular for unit vectors i, j and k along X,Y and Z axes. $i \times i = j \times j = k \times k = 0$ 4. **Vector product of perpendicular vectors:** If A and B are perpendicular vectors, then $\theta = 90°$ $A \times B = AB sin 90° n = AB n$ Thus, the vector product of two vectors has magnitude equal to the product of magnitudes of given vectors. If perpendicular vectors are unit vectors A and B, then $A × B = | A | | B | n = n$ In particular for unit vectors i, j, k, we have by right hand screw rule $i \times j = k$ $j \times i = -k$ $j \times k = i$ $k \times j = -i$ $k \times i = j$ $i \times k = -j$.