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ABCD is a rectangle. M is the midpoint of CD. Find the following in terms of u and v.

August 11, 2026Mathematics

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2 parts

-\mathbf{u} + \mathbf{v}

-\mathbf{u} + \mathbf{w}

4 steps

let's continue with Questions 2 and 3.

For Questions 2 and 3, refer to the new diagram where:
$\vec{AB} = \mathbf{u}$
$\vec{AD} = \mathbf{v}$
$\vec{AC} = \mathbf{w}$

Express $\vec{BD}$ in terms of $\mathbf{u}$ , $\mathbf{v}$ , and $\mathbf{w}$ .

Use the triangle rule for vector addition.
To go from B to D, we can follow the path from B to A, then from A to D.
$\vec{BD} = \vec{BA} + \vec{AD}$
Substitute the given vectors.
We know $\vec{BA} = -\vec{AB}$ , so $\vec{BA} = -\mathbf{u}$ .
We are given $\vec{AD} = \mathbf{v}$ .
$\vec{BD} = -\mathbf{u} + \mathbf{v}$
The expression for $\vec{BD}$ is $\boxed{-\mathbf{u} + \mathbf{v}}$ .

Express $\vec{BC}$ in terms of $\mathbf{u}$ , $\mathbf{v}$ , and $\mathbf{w}$ .

Use the triangle rule for vector addition.
To go from B to C, we can follow the path from B to A, then from A to C.
$\vec{BC} = \vec{BA} + \vec{AC}$
Substitute the given vectors.
We know $\vec{BA} = -\vec{AB}$ , so $\vec{BA} = -\mathbf{u}$ .
We are given $\vec{AC} = \mathbf{w}$ .
$\vec{BC} = -\mathbf{u} + \mathbf{w}$
The expression for $\vec{BC}$ is $\boxed{-\mathbf{u} + \mathbf{w}}$ .

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