Question 1: What does a vector represent in physics?
A) A physical quantity with only magnitude
B) A physical quantity with both magnitude and direction
C) A scalar quantity
D) A quantity with only direction
Explanation: A vector is a quantity that has both magnitude and direction, such as force, velocity, and displacement.
Question 2: Which of the following is an example of a vector quantity?
A) Temperature
B) Mass
C) Displacement
D) Time
Explanation: Displacement is a vector quantity because it has both magnitude and direction.
Question 3: What is the unit of a vector in physics?
A) Meter (m)
B) Kilogram (kg)
C) There is no unit for a vector, only for its magnitude
D) Second (s)
Explanation: A vector does not have a specific unit; the unit corresponds to the physical quantity it represents, such as meter for displacement or newton for force.
Question 4: What is the result of adding two vectors in the same direction?
A) A vector with zero magnitude
B) A vector whose magnitude is the sum of the magnitudes of the two vectors
C) A vector whose magnitude is the difference between the magnitudes of the two vectors
D) A vector that points in a direction perpendicular to both
Explanation: When two vectors are added in the same direction, their magnitudes add up to form a new vector whose magnitude is the sum of the individual magnitudes.
Question 5: What is the vector sum of two vectors in the opposite directions?
A) A vector with a magnitude equal to the sum of the vectors
B) A vector with a magnitude equal to the difference of the magnitudes of the vectors
C) A vector with zero magnitude
D) A vector pointing perpendicular to both vectors
Explanation: When two vectors are in opposite directions, the resultant vector's magnitude is the difference between the magnitudes of the two vectors.
Question 6: Which operation gives the magnitude of the resultant vector of two vectors?
A) Dot product
B) Cross product
C) Scalar multiplication
D) Vector addition
Explanation: The cross product of two vectors results in a vector perpendicular to both, and its magnitude can be used to calculate the area of the parallelogram formed by the vectors.
Question 7: What is the dot product of two perpendicular vectors?
A) Zero
B) The product of their magnitudes
C) The sum of their magnitudes
D) The magnitude of their cross product
Explanation: The dot product of two perpendicular vectors is zero, as the cosine of 90° is zero.
Question 8: Which of the following is the correct unit for vector multiplication?
A) Meter
B) Newton
C) There is no unit, it depends on the vectors multiplied
D) Joule
Explanation: The units of vector multiplication depend on the physical quantities involved. For instance, the dot product results in a scalar quantity with units like joules or newtons, while the cross product results in a vector with units like newton-meters.
Question 9: What does the magnitude of the cross product of two vectors represent?
A) The area of the parallelogram formed by the vectors
B) The angle between the vectors
C) The length of the vector
D) The scalar product of the vectors
Explanation: The magnitude of the cross product of two vectors gives the area of the parallelogram formed by them.
Question 10: If two vectors are collinear and have the same direction, what will be their resultant vector?
A) The sum of their magnitudes in the same direction
B) A vector with zero magnitude
C) The difference of their magnitudes in the same direction
D) A vector perpendicular to both
Explanation: When two collinear vectors have the same direction, the resultant vector is the sum of their magnitudes and points in the same direction.
Question 11: Which of the following is true for two vectors that are parallel?
A) The dot product of the vectors is positive or zero
B) The cross product of the vectors is non-zero
C) The vectors are always perpendicular
D) The vectors have no magnitude
Explanation: The dot product of two parallel vectors is positive or zero, and their cross product is zero, as they are either in the same or opposite direction.
Question 12: What is the vector sum of three vectors A, B, and C if they are coplanar and form a triangle?
A) The sum is equal to zero
B) The sum is equal to the magnitude of the resultant vector
C) The sum is equal to the difference between the largest and smallest vectors
D) The sum is not defined in this case
Explanation: When three vectors form a triangle, their vector sum is zero, as they form a closed loop.
Question 13: What is the component of vector A along the direction of vector B?
A) A × B
B) A cos(θ), where θ is the angle between A and B
C) A × sin(θ)
D) A sin(θ), where θ is the angle between A and B
Explanation: The component of vector A along the direction of vector B is given by A cos(θ), where θ is the angle between the two vectors.
Question 14: What happens when two vectors are added using the triangle law of vector addition?
A) The resultant vector is represented by the third side of the triangle
B) The resultant vector is the sum of their magnitudes
C) The resultant vector is perpendicular to both vectors
D) The vectors cancel each other out
Explanation: The triangle law of vector addition states that if two vectors are represented as two sides of a triangle, then the third side gives the magnitude and direction of the resultant vector.
Question 15: How is the direction of a vector represented in 3D space?
A) Using three coordinates: x, y, and z
B) Using two coordinates: x and y
C) Using polar coordinates
D) Using only one coordinate: x
Explanation: In 3D space, a vector's direction is represented by three coordinates: x, y, and z, which describe its position relative to the origin.
Question 16: What is the result when the cross product of two parallel vectors is computed?
A) A vector in the same direction as the two vectors
B) Zero vector
C) A scalar value
D) A vector perpendicular to both vectors
Explanation: The cross product of two parallel vectors is zero because there is no perpendicular component between them.
Question 17: Which of the following vectors is orthogonal to both vector A and vector B?
A) The sum of vectors A and B
B) The difference of vectors A and B
C) The cross product of vectors A and B
D) The dot product of vectors A and B
Explanation: The cross product of two vectors results in a vector that is perpendicular (orthogonal) to both vectors.
Question 18: What does the scalar product (dot product) of two vectors give?
A) A scalar quantity
B) A vector quantity
C) The area of the parallelogram formed by the vectors
D) The perpendicular distance between the vectors
Explanation: The dot product of two vectors results in a scalar quantity, which is calculated as the product of their magnitudes and the cosine of the angle between them.
Question 19: Which of the following operations involves the dot product of vectors?
A) Calculating the area of a triangle
B) Determining the perpendicular distance between vectors
C) Finding the projection of one vector onto another
D) Calculating the resultant of two vectors
Explanation: The dot product is used to find the projection of one vector onto another, as it relates to the cosine of the angle between the vectors.
Question 20: If two vectors have the same magnitude but different directions, what is their resultant?
A) The magnitude depends on the angle between them
B) The resultant is always zero
C) The resultant is equal to the sum of the magnitudes
D) The resultant is a vector perpendicular to both
Explanation: The resultant of two vectors with the same magnitude but different directions depends on the angle between them and can be calculated using the law of cosines or the parallelogram law.
Question 21: What is the condition for two vectors to be perpendicular?
A) The angle between them must be 90°
B) Their dot product must be zero
C) The magnitude of their cross product must be zero
D) The magnitude of both vectors must be the same
Explanation: Two vectors are perpendicular if the angle between them is 90°, which results in their dot product being zero.
Question 22: Which operation results in a vector that is perpendicular to the plane formed by two vectors?
A) The cross product of the two vectors
B) The dot product of the two vectors
C) The scalar multiplication of the two vectors
D) The sum of the two vectors
Explanation: The cross product of two vectors gives a vector that is perpendicular to both of the original vectors and lies in the plane formed by them.
Question 23: What is the result when you add two vectors of the same magnitude but opposite directions?
A) The resultant vector is zero
B) The resultant vector has twice the magnitude of the original vectors
C) The resultant vector has half the magnitude of the original vectors
D) The resultant vector points in the direction of the larger vector
Explanation: When two vectors of equal magnitude but opposite directions are added, they cancel each other out, resulting in a zero vector.
Question 24: What does the vector A × B (cross product) represent geometrically?
A) The area of the parallelogram formed by vectors A and B
B) The angle between vectors A and B
C) The projection of vector A onto vector B
D) The magnitude of the resultant of vectors A and B
Explanation: The magnitude of the cross product of two vectors represents the area of the parallelogram formed by them.
Question 25: If the vectors A = (3, -2, 1) and B = (1, 0, 4), what is the cross product A × B?
A) (6, -11, -2)
B) (8, -11, 2)
C) (-8, 11, -2)
D) (3, 0, 4)
Explanation: The cross product of vectors A = (3, -2, 1) and B = (1, 0, 4) is calculated as (3, -2, 1) × (1, 0, 4) = (8, -11, 2).
Question 26: Which of the following is true about the direction of the cross product of two vectors?
A) It is along the line joining the tail of the vectors
B) It is perpendicular to both vectors
C) It is in the same direction as one of the original vectors
D) It is parallel to the plane formed by the two vectors
Explanation: The direction of the cross product of two vectors is perpendicular to the plane formed by the two vectors.
Question 27: What is the magnitude of the cross product of two vectors A and B?
A) |A| |B| sin(θ)
B) |A| |B| cos(θ)
C) |A| + |B|
D) |A| - |B|
Explanation: The magnitude of the cross product of two vectors is given by |A| |B| sin(θ), where θ is the angle between the vectors.
Question 28: Which of the following is not a property of vector addition?
A) Commutativity
B) Distributivity over scalar multiplication
C) Associativity
D) Closure under addition
Explanation: Vector addition satisfies commutativity, associativity, and closure under addition, but distributivity over scalar multiplication is a property of scalar multiplication, not addition.
Question 29: The dot product of two vectors gives the projection of one vector onto the other. Which of the following is the correct formula for the dot product?
A) |A| × |B|
B) |A| |B| cos(θ)
C) |A| |B| sin(θ)
D) |A| + |B|
Explanation: The dot product of two vectors is given by |A| |B| cos(θ), where θ is the angle between them.
Question 30: What is the result of the dot product of two perpendicular vectors?
A) Zero
B) One
C) The product of their magnitudes
D) The sum of their magnitudes
Explanation: The dot product of two perpendicular vectors is zero, as the cosine of 90° is zero.
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