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Questions and Answers
Which of the following is a scalar quantity?
Which of the following is a scalar quantity?
What denotes the modulus of a vector?
What denotes the modulus of a vector?
Which type of vector has a modulus equal to zero?
Which type of vector has a modulus equal to zero?
What is true about equal vectors?
What is true about equal vectors?
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Which of the following vectors is represented by a directed line segment with an arrow?
Which of the following vectors is represented by a directed line segment with an arrow?
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What characterizes co-planar vectors?
What characterizes co-planar vectors?
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If two vectors have the same magnitude but opposite directions, they are referred to as what?
If two vectors have the same magnitude but opposite directions, they are referred to as what?
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Which of the following vectors can be classified as a proper vector?
Which of the following vectors can be classified as a proper vector?
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What is the value of the dot product of two perpendicular vectors?
What is the value of the dot product of two perpendicular vectors?
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What does the dot product of a vector with itself equal?
What does the dot product of a vector with itself equal?
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Which of the following statements about the dot product is true?
Which of the following statements about the dot product is true?
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What characteristic defines two vectors as orthogonal vectors?
What characteristic defines two vectors as orthogonal vectors?
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What is the result of the cross product of two parallel vectors?
What is the result of the cross product of two parallel vectors?
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If the angle between two vectors is 90°, what is the magnitude of their cross product?
If the angle between two vectors is 90°, what is the magnitude of their cross product?
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Which of the following correctly describes the addition of two vectors using the head-and-tail rule?
Which of the following correctly describes the addition of two vectors using the head-and-tail rule?
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What does the resultant of several vectors in a closed figure equal?
What does the resultant of several vectors in a closed figure equal?
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What property does the cross product of two vectors exhibit?
What property does the cross product of two vectors exhibit?
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What is the result when calculating the cross product of a vector with itself?
What is the result when calculating the cross product of a vector with itself?
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Which law states that the order in which vectors are added does not affect the resultant?
Which law states that the order in which vectors are added does not affect the resultant?
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What does the dot product of two orthogonal unit vectors yield?
What does the dot product of two orthogonal unit vectors yield?
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What does subtracting vector b from vector a involve?
What does subtracting vector b from vector a involve?
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What remains unchanged when a vector is multiplied by a scalar?
What remains unchanged when a vector is multiplied by a scalar?
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Which of the following results in a scalar product of two vectors equal to zero?
Which of the following results in a scalar product of two vectors equal to zero?
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What is the maximum value of the dot product of two vectors dependent on?
What is the maximum value of the dot product of two vectors dependent on?
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What is the formula to calculate the x-component of the position vector r?
What is the formula to calculate the x-component of the position vector r?
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When two vectors are in opposite directions, how is net displacement determined?
When two vectors are in opposite directions, how is net displacement determined?
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If two vectors are perpendicular to each other, what method is used to find the net displacement?
If two vectors are perpendicular to each other, what method is used to find the net displacement?
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What is the magnitude of the resultant when two vectors of 10m are at an angle of 60° to each other?
What is the magnitude of the resultant when two vectors of 10m are at an angle of 60° to each other?
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What is the direction of the resultant when calculating two vectors of equal length at an angle of 60°?
What is the direction of the resultant when calculating two vectors of equal length at an angle of 60°?
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What is the formula for the cross product of two vectors a and b?
What is the formula for the cross product of two vectors a and b?
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How is the direction of the resultant vector determined when using the Triangle Law of Vector Addition?
How is the direction of the resultant vector determined when using the Triangle Law of Vector Addition?
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What is the magnitude of the resultant vector c when the angle θ is 0 degrees?
What is the magnitude of the resultant vector c when the angle θ is 0 degrees?
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What happens to the resultant vector c when the angle θ is 180 degrees?
What happens to the resultant vector c when the angle θ is 180 degrees?
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In resolving a vector into its perpendicular components, what does point P represent?
In resolving a vector into its perpendicular components, what does point P represent?
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Which case of the Triangle Law of Vector Addition leads to the maximum resultant magnitude?
Which case of the Triangle Law of Vector Addition leads to the maximum resultant magnitude?
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Which expression correctly represents the direction of resultant vector c for θ = 90 degrees?
Which expression correctly represents the direction of resultant vector c for θ = 90 degrees?
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What is the resultant vector when both vectors a and b are equal and θ = 180 degrees?
What is the resultant vector when both vectors a and b are equal and θ = 180 degrees?
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Study Notes
Scalars and Vectors
- Scalars: Quantities with only magnitude (e.g., mass, time, distance, speed).
- Vectors: Quantities with both magnitude and direction (e.g., displacement, velocity, acceleration, force).
Vector Notation and Representation
- Vectors denoted by an alphabet with an arrow (→) over its head (e.g., velocity is denoted as V→).
- Represented by a directed line segment:
- Length indicates magnitude.
- Arrow indicates direction.
Vector Modulus
- Modulus represents only the magnitude of a vector.
- Denoted by enclosing the vector within two vertical parallel lines (e.g., |a|).
- The modulus of a vector 'a' is: |a| = √a² = a.
Vector Classification
- Unit Vector: Modulus equals 1 (denoted by a small letter with a caret or hat symbol).
- Null Vector: Modulus equals zero. Example: The velocity of a body vertically projected upwards at its maximum height.
- Proper Vector: Modulus is non-zero.
- Like Vectors: Same direction but different magnitudes.
- Unlike Vectors: Opposite direction and different magnitudes.
- Equal Vectors: Same magnitude and direction.
- (-ve) Vectors: Same magnitude, opposite directions.
- Co-linear Vectors: All vectors lie on the same line.
- Co-planner Vectors: All vectors lie in the same plane.
- Co-initial Vectors: All vectors have the same origin.
- Orthogonal Vectors: Vectors perpendicular to each other.
Vector Addition
- Head-and-tail rule: Joining the head of one vector with the tail of another results in the resultant vector.
- General Addition: AB + BC = AC, PB + BR = PR. You cannot add vectors if they are not head-to-tail.
- Addition of Several Vectors: The resultant is found by connecting the tail of the first vector to the head of the last vector. The resultant of vectors in a closed figure is zero.
Vector Addition Properties
- Commutative Law: a + b = b + a.
- Associative Law: (a + b) + c = a + (b + c).
Vector Subtraction
- Subtracting vector 'b' from 'a' is the same as adding -b to a (a-b = a + (-b)).
- To find -b, reverse the direction of b while keeping the same magnitude.
Vector Multiplication
- Scalar Product (Dot Product): a.b = |a| |b| cos θ = ab cos θ.
- Vector Product (Cross Product): a x b = |a| |b| sin θ n = ab sin θ n.
Scalar Product Properties
- Maximum value when θ = 0° (vectors in the same direction).
- Value is 0 when θ = 90° (vectors perpendicular).
- Dot product of a vector with itself: a.a = |a|² = a².
- Commutative: a.b = b.a.
- Dot product of unit vectors: î.î = j.j = k.k = 1; î.j = j.k = k.i = 0.
Vector Product Properties
- |a x b| = ab sin θ.
- Maximum value when θ = 90° (vectors perpendicular).
- Value is 0 when θ = 0° (vectors in the same direction).
- Cross product of a vector with itself is 0: a x a = 0.
- Anti-commutative: (a x b) = -(b x a).
Triangle Law of Vector Addition
- If two vectors are represented by sides of a triangle, their resultant is represented by the third side, but the direction is reversed.
- Resultant magnitude: c = √(a² + b² + 2ab cos θ).
- Resultant direction: α = tan-1 (b sin θ/a + b cos θ).
Special Triangle Law Cases
- θ = 0°: Vectors in the same direction, |c| = |a| + |b|.
- θ = 180°: Vectors in opposite directions, |c| = |a| - |b|.
- θ = 90°: Vectors perpendicular, use Pythagorean theorem to find the resultant.
Resolving a Vector into Components
- A vector can be resolved into two perpendicular components.
- For a vector r = xi + yj, where x = r cos θ and y = r sin θ, the vector can be written as: r = r cos θ i + r sin θ j.
Examples
- Same Direction: A car travelling 200m East and 300m East has a net displacement of 500m East.
- Opposite Directions: A car travelling 200m East and 300m West has a net displacement of 100m West.
- Perpendicular Vectors: An object moving 10m North and 10m East has a net displacement of 14.1m.
- Angle between Vectors: Two vectors of 10m each, with an angle of 60° between them, have a resultant magnitude of 10√3m and a direction of 30°.
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Description
Test your understanding of scalars and vectors with this quiz. Dive into key concepts such as vector notation, representation, and classification. Challenge yourself to identify different types of vectors and their properties.
