What is the magnitude of 0, -7? [1 point]

College Board AP 2025 (style)-style practice: Section II (free response, no calculator). Let u = 2, 5 and v = -1, 3. (a) Find u + v and 3u. (b) Find the magnitude of u + v.

A 4-point question on vector operations. (a) Operations (2 points): add componentwise, u + v = 2 + (-1), 5 + 3 = 1, 8. Scale each component, 3u = 3 · 2, 3 · 5 = 6, 15. (b) Magnitude (2 points): |u + v| = | 1, 8 | = sqrt(1^2 + 8^2) = sqrt(1 + 64) = sqrt(65).

United StatesPrecalculusSyllabus dot point

What is a vector, and how do you add, scale and find the magnitude and direction of one?

Topic 4.8 Vectors: represent a vector by components, compute its magnitude and direction, and add, subtract and scale vectors.

A focused answer to AP Precalculus Topic 4.8, covering vectors as objects with magnitude and direction, component form, magnitude and direction angle, scalar multiplication, and vector addition and subtraction.

Generated by Claude Opus 4.89 min answerUpdated 2026-06-04

Reviewed by: AI editorial process; not yet individually human-reviewed

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Quick answer

A vector is a quantity with both magnitude (size) and direction, unlike a scalar, which has only size. In component form a vector is written $\langle a, b \rangle$ , where $a$ is the horizontal change and $b$ the vertical change from tail to head. Its magnitude is $|\langle a, b \rangle| = \sqrt{a^2 + b^2}$ , the length by Pythagoras, and its direction angle is $\theta = \arctan\frac{b}{a}$ , adjusted for the quadrant. To scale a vector, multiply each component by the scalar, which stretches or shrinks it (and reverses it if the scalar is negative). To add or subtract vectors, combine corresponding components: $\langle a, b \rangle + \langle c, d \rangle = \langle a + c, b + d \rangle$ . Geometrically, addition is the tip-to-tail rule, and scaling changes length without changing the line of direction (unless negated).

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What this topic is asking
What a vector is
Magnitude and direction
Scaling, adding and subtracting
Why components make this easy
Try this

What this topic is asking

The College Board (Topic 4.8) wants you to work with vectors: quantities with both magnitude and direction. You represent a vector in component form, compute its magnitude and direction angle, and perform the basic operations of scalar multiplication, addition and subtraction.

What a vector is

A scalar, by contrast, is a single number with size only. The component form turns the geometric arrow into algebra, so operations become arithmetic on the components.

Magnitude and direction

The magnitude is always non-negative, and the direction angle needs the same quadrant adjustment as the polar conversion of Topic 3.13, since arctangent alone cannot distinguish opposite directions.

Scaling, adding and subtracting

Combining vectors and finding direction

Let $\mathbf{u} = \langle 4, 3 \rangle$ and $\mathbf{v} = \langle -1, 1 \rangle$ . Find $2\mathbf{u} - \mathbf{v}$ , its magnitude, and its direction angle.

step 1 Scale u

$2\mathbf{u} = \langle 2 \cdot 4, 2 \cdot 3 \rangle = \langle 8, 6 \rangle$ .

step 2 Subtract v componentwise

$2\mathbf{u} - \mathbf{v} = \langle 8 - (-1), 6 - 1 \rangle = \langle 9, 5 \rangle$ .

step 3 Find the magnitude

$|\langle 9, 5 \rangle| = \sqrt{9^2 + 5^2} = \sqrt{81 + 25} = \sqrt{106}$ .

step 4 Find the direction angle

$\theta = \arctan\frac{5}{9} \approx 29.1^\circ$ . Both components are positive (Quadrant I), so no adjustment is needed; the vector points up and to the right at about $29^\circ$ above the horizontal.

Why components make this easy

A point worth stating once is that component form reduces every vector operation to arithmetic done separately on the horizontal and vertical parts. Addition, subtraction and scaling all act componentwise, which is why a vector behaves like the position model of Topic 4.2: the $x$ -part and $y$ -part evolve independently. The magnitude and direction then repackage the components into the polar description "how long and which way", connecting vectors back to the trigonometry of Unit 3. Treating components and magnitude-direction as two views of the same object, and converting freely between them, is the core fluency this topic builds toward the vector-valued functions of Topic 4.9.

Try this

Q1. Find $\langle 1, 2 \rangle + \langle 3, -5 \rangle$ . [1 point]

Cue. Add components: $\langle 1 + 3, 2 + (-5) \rangle = \langle 4, -3 \rangle$ .

Q2. What is the magnitude of $\langle 0, -7 \rangle$ ? [1 point]

Cue. $\sqrt{0^2 + (-7)^2} = \sqrt{49} = 7$ .

Exam-style practice questions

Practice questions written in the style of College Board exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

AP 2025 (style)1 marksSection I, Part A (multiple choice, no calculator). What is the magnitude of the vector

\langle 3, -4 \rangle

? (A)

1

(B)

5

(C)

7

(D)

\sqrt{7}

Show worked answer →

The correct answer is (B), $5$ .

The magnitude of $\langle a, b \rangle$ is $\sqrt{a^2 + b^2}$ . Here $\sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$ . The signs of the components do not matter once they are squared.

AP 2025 (style)4 marksSection II (free response, no calculator). Let

\mathbf{u} = \langle 2, 5 \rangle

and

\mathbf{v} = \langle -1, 3 \rangle

. (a) Find

\mathbf{u} + \mathbf{v}

and

3\mathbf{u}

. (b) Find the magnitude of

\mathbf{u} + \mathbf{v}

Show worked answer →

A 4-point question on vector operations.

(a) Operations (2 points): add componentwise, $\mathbf{u} + \mathbf{v} = \langle 2 + (-1), 5 + 3 \rangle = \langle 1, 8 \rangle$ . Scale each component, $3\mathbf{u} = \langle 3 \cdot 2, 3 \cdot 5 \rangle = \langle 6, 15 \rangle$ .
(b) Magnitude (2 points): $|\mathbf{u} + \mathbf{v}| = |\langle 1, 8 \rangle| = \sqrt{1^2 + 8^2} = \sqrt{1 + 64} = \sqrt{65}$ .

Related dot points

Sources & how we know this

AP Precalculus Course and Exam Description — College Board (2023)