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Ch-10: Vector Algebra
12 Questions
Before You Practice Try each question on your own first. Use the Show Answer button only after you have selected your option, then read the explanation and revise the related concept from your chapter notes.
1. A vector quantity is characterized by having:
A. Both magnitude and direction
B. Only magnitude
C. Only direction
D. Neither magnitude nor direction
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Answer: A. Both magnitude and direction
Explanation: Unlike a scalar, a vector quantity requires both a numerical magnitude and an associated direction to be fully specified.
2. The magnitude of a vector a = xi + yj + zk is given by:
A. √(x² + y² + z²)
B. x + y + z
C. xyz
D. x² + y² + z²
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Answer: A. √(x² + y² + z²)
Explanation: This follows from a direct extension of the Pythagorean theorem to three dimensions.
3. Two vectors are said to be equal if they have:
A. The same magnitude and the same direction
B. Only the same magnitude
C. Only the same direction
D. The same initial point only
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Answer: A. The same magnitude and the same direction
Explanation: Vector equality depends solely on magnitude and direction, not on the particular location of the vector.
4. The dot (scalar) product of two vectors a and b is given by:
A. |a||b| cos θ
B. |a||b| sin θ
C. |a| + |b|
D. |a| - |b|
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Answer: A. |a||b| cos θ
Explanation: The dot product depends on the cosine of the angle between the two vectors and their magnitudes.
5. The magnitude of the cross (vector) product of two vectors a and b is given by:
A. |a||b| sin θ
B. |a||b| cos θ
C. |a| + |b|
D. |a| · |b| always
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Answer: A. |a||b| sin θ
Explanation: Unlike the dot product, the cross product's magnitude depends on the sine of the angle between the vectors.
6. Two vectors are said to be perpendicular if their dot product equals:
A. Zero
B. One
C. The product of their magnitudes
D. An undefined value
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Answer: A. Zero
Explanation: Since cos 90° = 0, a zero dot product indicates that the angle between the vectors is 90°.
7. Two non-zero vectors are said to be parallel (collinear) if their cross product is:
A. The zero vector
B. Equal to their dot product
C. At its maximum possible value
D. Undefined
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Answer: A. The zero vector
Explanation: Since sin 0° = sin 180° = 0, a zero cross product indicates the vectors are parallel or anti-parallel.
8. A unit vector is defined as a vector whose magnitude is:
A. 1
B. 0
C. Equal to its own direction
D. Undefined
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Answer: A. 1
Explanation: By definition, a unit vector has a magnitude of exactly one and is used to indicate pure direction.
9. The position vector of a point P(x, y, z) with respect to the origin is:
A. xi + yj + zk
B. x + y + z
C. Simply the coordinates (x, y, z)
D. |x| + |y| + |z|
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Answer: A. xi + yj + zk
Explanation: The position vector expresses the displacement of point P from the origin using the standard unit vectors i, j, and k.
10. The scalar triple product a · (b × c) geometrically represents:
A. The volume of the parallelepiped formed by the three vectors
B. The area of a triangle formed by the vectors
C. The angle between the vectors
D. The unit vector along a
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Answer: A. The volume of the parallelepiped formed by the three vectors
Explanation: The absolute value of the scalar triple product gives the volume of the parallelepiped with a, b, and c as coterminous edges.
11. The sum of two vectors a and b is found geometrically using the:
A. Triangle law or parallelogram law of vector addition
B. Simple multiplication rule
C. Cross product only
D. Division rule
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Answer: A. Triangle law or parallelogram law of vector addition
Explanation: Vector addition is performed geometrically by placing vectors head-to-tail (triangle law) or from a common point (parallelogram law).
12. The projection of vector a on vector b is given by:
A. (a · b) / |b|
B. (a × b) / |b|
C. |a||b|
D. a · b alone, without normalizing
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Answer: A. (a · b) / |b|
Explanation: The scalar projection of a onto b is obtained by dividing their dot product by the magnitude of b.
