Cross Product Practice Test

Advanced Algebra Practice Test: ACT math skills.

Cross Product Practice Test

This test has 20 questions

Instant feedback · Worked explanations

Vector Questions · Cross Product

Forge a vector out of the plane.

The cross product takes two three-component vectors and produces a new vector perpendicular to both. Its direction depends on order, while its magnitude measures the area created by the inputs.

This school-level review focuses on component calculation, the right-hand rule, order reversal, perpendicularity checks, coordinate basis patterns, angles, areas, and parallel vectors.

Two inputsThree matching components in each vector

OrientationOrder determines the perpendicular direction

Vector outputA normal vector with an area-based magnitude

Forge 01

The cross product creates a perpendicular vector

Unlike the dot product, the result is a vector rather than a scalar.

Input pair

In the standard school treatment, each input has three components.

u = ⟨ u_{1}, u_{2}, u_{3} ⟩, v = ⟨ v_{1}, v_{2}, v_{3} ⟩

Cross operation

Perpendicular output

The output is orthogonal to both inputs unless the cross product is the zero vector.

(u \times v) \cdot u = 0, (u \times v) \cdot v = 0

Direction

The right-hand rule selects one of the two perpendicular directions.

Magnitude

The length reflects the sizes of the inputs and the angle between them.

Zero case

Parallel inputs do not span an area, so their cross product is zero.

Forge 02

The right-hand rule fixes the output direction

Curl from the first vector toward the second through the smaller angle; the thumb gives the cross-product direction.

Order is not interchangeable

The two outputs have the same magnitude but opposite directions.

v \times u = - (u \times v)

Point fingers

Start along the first vector.

Curl toward second

Use the smaller angle between the inputs.

Read the thumb

The thumb points along the cross product.

Forge 03

Build the three output components with a fixed pattern

Each component is a difference of two products; the middle position is the most common source of sign errors.

u \times v = ⟨ u_{2} v_{3} - u_{3} v_{2}, u_{3} v_{1} - u_{1} v_{3}, u_{1} v_{2} - u_{2} v_{1} ⟩

First output component

u_{2} v_{3} - u_{3} v_{2}

Second output component

u_{3} v_{1} - u_{1} v_{3}

Third output component

u_{1} v_{2} - u_{2} v_{1}

Determinant layout

u \times v = | \begin{matrix} i & j & k \\ u_{1} & u_{2} & u_{3} \\ v_{1} & v_{2} & v_{3} \end{matrix} |

Expansion signs

+ i, - j, + k

The direct component formula already absorbs the middle negative sign.

Forge 04

Worked example: calculate each component separately

Keep the order fixed from the first input to the second.

Input vectors

u = ⟨ 1, 2, 3 ⟩, v = ⟨ 4, 5, 6 ⟩

Expected output

The answer must be a three-component vector. A scalar cannot be a cross-product result.

First

2 (6) - 3 (5) = 12 - 15

- 3

Second

3 (4) - 1 (6) = 12 - 6

6

Third

1 (5) - 2 (4) = 5 - 8

- 3

u \times v = ⟨ - 3, 6, - 3 ⟩

Forge 05

Verify the output with two dot products

A correct cross product is perpendicular to each original vector.

Candidate output

n = ⟨ - 3, 6, - 3 ⟩

Check first input

n \cdot u = - 3 + 12 - 9 = 0

Check second input

n \cdot v = - 12 + 30 - 18 = 0

n \cdot u = 0 and n \cdot v = 0 \Rightarrow perpendicular to both

Forge 06

The coordinate basis follows a three-step cycle

Moving forward around the cycle gives a positive result; moving backward gives its negative.

Memorize the cycle, not nine isolated facts

Repeat an axis with itself and the result is zero. Reverse any positive basis pair and the sign becomes negative.

i \times j = k

j \times k = i

k \times i = j

Forge 07

The magnitude depends on the sine of the angle

The output is largest for perpendicular inputs and zero for parallel inputs.

∥ u \times v ∥ = ∥ u ∥ ∥ v ∥ \sin θ

Input vectors

u = ⟨ 1, 0, 0 ⟩, v = ⟨ 1, 1, 0 ⟩

Cross magnitude

u \times v = ⟨ 0, 0, 1 ⟩ \Rightarrow ∥ u \times v ∥ = 1

Find the angle

1 = 1 \cdot \sqrt{2} \sin θ \Rightarrow θ = 45 °

Forge 08

Cross-product magnitude measures area

The parallelogram uses the full magnitude; a triangle with the same sides uses half.

Base times perpendicular height

The sine factor supplies the height relative to the chosen base.

A_{parallelogram} = ∥ u \times v ∥

Example sides

u = ⟨ 3, 0, 0 ⟩, v = ⟨ 1, 4, 0 ⟩

Parallelogram

u \times v = ⟨ 0, 0, 12 ⟩ \Rightarrow A = 12

Triangle

A = \frac{1}{2} (12) = 6

Forge 09

Recognize the zero-vector cases quickly

Parallel vectors, opposite vectors, and a repeated vector do not span a parallelogram.

Same direction

u \times (c u) = 0 for c > 0

The angle is zero, so the sine factor is zero.

Opposite direction

u \times (c u) = 0 for c < 0

The angle is straight, which also has sine zero.

Repeated vector

u \times u = 0

No distinct directions are available to create area.

Forge 10

Skills Covered

The problems combine signed arithmetic, three-dimensional components, geometry, and interpretation.

Calculate

Build all three cross-product components in the correct order.

Orient

Use order and the right-hand rule to select a direction.

Verify

Confirm perpendicularity with two dot products.

Interpret

Connect magnitude to angles, areas, and parallelism.

Forge 11

How to Approach the Test

Separate order, component arithmetic, and interpretation into distinct checks.

1Record orderWrite the first vector above the second before calculating.

2Build componentsUse the fixed difference-of-products pattern.

3Check signsInspect the middle component and any negative entries.

4Interpret outputVerify direction, perpendicularity, magnitude, area, or requested form.

Forge 12

Common Mistakes

Most errors come from reversing order, losing the middle sign, or confusing vector and scalar outputs.

Swapping the inputs silently

Cross products change sign when order reverses.

Write the requested order before any arithmetic.

Returning a scalar

The cross product has three components.

Place the three results inside vector brackets.

Using the same pair order everywhere

Each output position uses a different pair of component indices.

Follow the fixed component formula carefully.

Losing the middle sign

Determinant expansion follows a positive-negative-positive pattern.

Use the direct second-component formula or show the negative explicitly.

Skipping perpendicularity checks

A small arithmetic error may still look plausible.

Dot the output with both original vectors.

Using cosine for magnitude

Cross-product magnitude uses sine of the included angle.

Cosine belongs to the dot-product angle formula.

Final inspection

Final cross-product audit

Inspect order, components, orientation, perpendicularity, and requested meaning.

Spatial Output Check

A correct result has three components, follows the requested order, and is perpendicular to both nonparallel inputs.

Order · Build · Orient · Verify

Was the input order preserved?Reversal changes the output sign.

Were all three components calculated?Each is a difference of two products.

Was the middle sign handled correctly?Compare with the direct component pattern.

Does the direction fit the right-hand rule?Check orientation before choosing an answer.

Is the output perpendicular to both inputs?Two dot products should equal zero.

Does the answer match the requested quantity?Distinguish a vector, magnitude, angle, and area.

Use this free 20-question practice test for high school Advanced Algebra review, ACT-style skill practice, placement preparation, or classroom practice. You can retake the test without creating an account. The examples in this review block are illustrative and are not copies of the test questions.