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Exercises: Add and Subtract Vectors

Work through each section in order. Show your work where indicated.

Grade 9·21 problems·~30 min·Common Core Math - HS Number and Quantity·standard·hsn-vm-b-4

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Recall / Warm-Up

To add vectors $\mathbf{u} + \mathbf{w}$ using the head-to-tail method, which step is correct?

Place the tail of $\mathbf{w}$ at the tip of $\mathbf{u}$ ; draw the resultant from the tail of $\mathbf{u}$ to the tip of $\mathbf{w}$ .

Place the tail of $\mathbf{u}$ at the tip of $\mathbf{w}$ ; draw the resultant from the tail of $\mathbf{w}$ to the tip of $\mathbf{u}$ .

Place both vectors at the origin; the resultant is the average of $\mathbf{u}$ and $\mathbf{w}$ .

Draw $\mathbf{u}$ and $\mathbf{w}$ end-to-end in any order; direction does not matter.

Let $\mathbf{u} = \langle 5, -2 \rangle$ and $\mathbf{w} = \langle -3, 4 \rangle$ . Find the $x$ -component of $\mathbf{u} + \mathbf{w}$ .

Which statement about the magnitude of a vector sum is always true?

$|\mathbf{u} + \mathbf{w}| = |\mathbf{u}| + |\mathbf{w}|$

$|\mathbf{u} + \mathbf{w}| \leq |\mathbf{u}| + |\mathbf{w}|$

$|\mathbf{u} + \mathbf{w}| \geq |\mathbf{u}| + |\mathbf{w}|$

$|\mathbf{u} + \mathbf{w}| = |\mathbf{u}| - |\mathbf{w}|$

Fluency Practice

Vectors $\mathbf{u}$ and $\mathbf{w}$ are drawn on a grid with the tail of $\mathbf{w}$ placed at the tip of $\mathbf{u}$ . Which vector is the resultant $\mathbf{u} + \mathbf{w}$ ?

The vector from the tail of $\mathbf{u}$ to the tip of $\mathbf{w}$

The vector from the tip of $\mathbf{u}$ to the tip of $\mathbf{w}$

The vector from the tip of $\mathbf{w}$ to the tail of $\mathbf{u}$

The vector from the tip of $\mathbf{u}$ to the tail of $\mathbf{w}$

Two vectors $\mathbf{u}$ and $\mathbf{w}$ share the same tail. A student completes the parallelogram. Using the parallelogram rule, which vector is $\mathbf{u} + \mathbf{w}$ ?

The diagonal from the shared tail to the opposite corner of the parallelogram

The other diagonal, connecting the tips of $\mathbf{u}$ and $\mathbf{w}$

The vector that bisects the angle between $\mathbf{u}$ and $\mathbf{w}$

The vector with magnitude $|\mathbf{u}| + |\mathbf{w}|$ at 45° between the two vectors

Let $\mathbf{u} = \langle 3, 4 \rangle$ and $\mathbf{w} = \langle 1, -2 \rangle$ . Find $|\mathbf{u} + \mathbf{w}|$ . Round to the nearest hundredth.

Let $\mathbf{v} = \langle 6, -1 \rangle$ and $\mathbf{w} = \langle -2, 5 \rangle$ . Find the $y$ -component of $\mathbf{v} - \mathbf{w}$ .

Vector $\mathbf{u}$ has magnitude 10 at $0\degree$ and vector $\mathbf{w}$ has magnitude 10 at $90\degree$ . Find $|\mathbf{u} + \mathbf{w}|$ . Give an exact value.

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