Before you read

Answer first — see what you already know.

If $a = ⟨ 2, 5 ⟩$ and $b = ⟨ 3, - 1 ⟩$ , what is $a + b$ ?

A
$⟨ 5, 6 ⟩$
B
$⟨ - 1, 6 ⟩$
C
$⟨ 6, 4 ⟩$
D
$⟨ 5, 4 ⟩$

Check your answer

Answer:

Before you read

Answer first — see what you already know.

What is the magnitude of $v = ⟨ 6, 8 ⟩$ ?

A
$100$
B
$14$
C
$14$
D
$10$

Check your answer

Answer:

Theory

What Is a Vector?

A vector has both magnitude (length) and direction. Written as $\vec{v} = \langle a, b \rangle$ or $\begin{pmatrix} a \\ b \end{pmatrix}$ .

Magnitude: $|\vec{v}| = \sqrt{a^2 + b^2}$

Operations:
- Addition: $\langle a, b \rangle + \langle c, d \rangle = \langle a+c, b+d \rangle$
- Subtraction: $\langle a, b \rangle - \langle c, d \rangle = \langle a-c, b-d \rangle$
- Scalar multiplication: $k \langle a, b \rangle = \langle ka, kb \rangle$

Vector addition: u + v = (3+1, 2+4) = (4, 6)

Example 1

If $\vec{u} = \langle 3, 4 \rangle$ and $\vec{v} = \langle -1, 2 \rangle$ , find $\vec{u} + \vec{v}$ and $|\vec{u}|$ .

$\vec{u} + \vec{v} = \langle 3+(-1), 4+2 \rangle = \langle 2, 6 \rangle$

$|\vec{u}| = \sqrt{9 + 16} = 5$

\vec{u} + \vec{v} = \langle 2, 6 \rangle

|\vec{u}| = 5

Tip

ACT Pro Tip

Vector questions on the ACT are usually straightforward component arithmetic. Treat the $x$ and $y$ components separately — it's just two parallel calculations. The magnitude is always the Pythagorean theorem.

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