GRE Physics Companion: Scalars and Vectors in Mechanics

This entry is the GRE-focused companion to M00-04: Scalars and Vectors in Mechanics. It assumes the vector algebra has already been learned and concentrates on rapid identification of the correct vector operation under time pressure.

1 Fast vector triage

Before calculating, ask what the problem is requesting:

• a magnitude or direction: resolve into components;
• a resultant: add components;
• a parallel projection, angle, or work: use a dot product;
• a perpendicular tendency, torque, or angular momentum: use a cross product;
• a direction only: normalize the vector.

2 Dot product versus cross product

The dot product is a scalar:

(1)

It is largest in magnitude when the vectors are parallel or antiparallel and vanishes when they are perpendicular.

The cross product is an axial vector:

(2)

It vanishes for parallel vectors and is largest when the vectors are perpendicular.

3 Worked GRE example 1: work without resolving every component

A force of magnitude 20 N acts at 60^∘ to a displacement of magnitude 3 m. The work is

(3)

so

(4)

The key recognition is that work is a dot product; no Cartesian decomposition is required.

4 Worked GRE example 2: identify when torque vanishes

A force acts along the line joining a pivot to its point of application. Since the angle between r and F is either 0^∘ or 180^∘,

(5)

The fastest route is recognizing the cross-product geometry rather than writing components.

5 GRE-speed questions

MECH-GRE-VEC-001

Which quantity is a vector?

1. kinetic energy
2. speed
3. momentum
4. work

MECH-GRE-VEC-002

A vector of magnitude 10 points 60^∘ above the positive x axis. Its x component is

1. 5
2. 5
3. 10
4. 10

MECH-GRE-VEC-003

If two nonzero vectors satisfy

(6)

then the vectors are

1. parallel
2. antiparallel
3. perpendicular
4. equal in magnitude

MECH-GRE-VEC-004

The magnitude of A × B is largest when the angle between A and B is

1. 0^∘
2. 30^∘
3. 60^∘
4. 90^∘

MECH-GRE-VEC-005

A force is perpendicular to a displacement. The work done by that force is

1. positive
2. negative
3. zero
4. equal to FΔr

MECH-GRE-VEC-006

A force acts directly along the position vector from a pivot to its point of application. The torque about the pivot is

1. maximum
2. zero
3. equal to rF
4. impossible to determine

MECH-GRE-VEC-007

A particle has momentum 3e_x + 4e_y in SI units. Its momentum magnitude is

1. 3
2. 4
3. 5
4. 7

MECH-GRE-VEC-008

If A = 2e_x and B = 3e_y, then A × B points in the

1. +x direction
2. +y direction
3. +z direction
4. −z direction

MECH-GRE-VEC-009

A vector A is multiplied by −2. Which statement is correct?

1. its magnitude halves and direction is unchanged
2. its magnitude doubles and direction reverses
3. its magnitude is unchanged and direction reverses
4. its magnitude doubles and direction is unchanged

MECH-GRE-VEC-010

A force F and displacement Δr satisfy F ⋅ Δr < 0. The angle between them must be

1. less than 90^∘
2. exactly 90^∘
3. greater than 90^∘ but less than or equal to 180^∘
4. zero

6 Answer key and brief rationale

1. C. Momentum has both magnitude and direction.
2. A. 10 cos 60^∘ = 5.
3. C. A zero dot product means perpendicular nonzero vectors.
4. D. sin 𝜃 is largest at 90^∘.
5. C. The dot product vanishes.
6. B. The cross product vanishes for parallel vectors.
7. C. = 5.
8. C. e_x × e_y = e_z.
9. B. A negative scalar reverses direction and the factor 2 doubles magnitude.
10. C. Negative dot product means cos 𝜃 < 0.

7 Use with the main article

Use this entry after M00-04. The main article develops vector concepts and mechanics applications in detail; this companion isolates timed recognition, elimination, and GRE-style response patterns.

References