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Given vectors a = (2, 3) and c = (3, 5), solve for vector b in the equation a + 2b = c, and then find the magnitude of (a + b).

April 22, 2026Mathematics

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2 parts

\begin{pmatrix} 0.5 \\ 1 \end{pmatrix}

4.72

7 steps

Here's the solution to your vector problem:

Find $\mathbf{b}$ .

Substitute the given vectors into the equation $\mathbf{a} + 2\mathbf{b} = \mathbf{c}$ .
$\begin{pmatrix} 2 \\ 3 \end{pmatrix} + 2\mathbf{b} = \begin{pmatrix} 3 \\ 5 \end{pmatrix}$
Isolate $2\mathbf{b}$ by subtracting vector $\mathbf{a}$ from both sides.
$2\mathbf{b} = \begin{pmatrix} 3 \\ 5 \end{pmatrix} - \begin{pmatrix} 2 \\ 3 \end{pmatrix}$
Perform the vector subtraction.
$2\mathbf{b} = \begin{pmatrix} 3 - 2 \\ 5 - 3 \end{pmatrix}$ $2\mathbf{b} = \begin{pmatrix} 1 \\ 2 \end{pmatrix}$
Solve for $\mathbf{b}$ by dividing each component by 2.
$\mathbf{b} = \frac{1}{2} \begin{pmatrix} 1 \\ 2 \end{pmatrix}$ $\mathbf{b} = \begin{pmatrix} \frac{1}{2} \\ \frac{2}{2} \end{pmatrix}$ $\mathbf{b} = \begin{pmatrix} 0.5 \\ 1 \end{pmatrix}$
The vector $\mathbf{b}$ is $\boxed{\begin{pmatrix} 0.5 \\ 1 \end{pmatrix}}$ .

Find the magnitude of $(\mathbf{a} + \mathbf{b})$ correct to 2 decimal places.

Calculate the vector $\mathbf{a} + \mathbf{b}$ .
$\mathbf{a} + \mathbf{b} = \begin{pmatrix} 2 \\ 3 \end{pmatrix} + \begin{pmatrix} 0.5 \\ 1 \end{pmatrix}$ $\mathbf{a} + \mathbf{b} = \begin{pmatrix} 2 + 0.5 \\ 3 + 1 \end{pmatrix}$ $\mathbf{a} + \mathbf{b} = \begin{pmatrix} 2.5 \\ 4 \end{pmatrix}$
Calculate the magnitude of $\mathbf{a} + \mathbf{b}$ .
The magnitude of a vector $\begin{pmatrix} x \\ y \end{pmatrix}$ is $\sqrt{x^2 + y^2}$ .
$|\mathbf{a} + \mathbf{b}| = \sqrt{(2.5)^2 + (4)^2}$ $|\mathbf{a} + \mathbf{b}| = \sqrt{6.25 + 16}$ $|\mathbf{a} + \mathbf{b}| = \sqrt{22.25}$
Calculate the square root and round to 2 decimal places.
$|\mathbf{a} + \mathbf{b}| \approx 4.7170$
Rounding to 2 decimal places:
$|\mathbf{a} + \mathbf{b}| \approx 4.72$
The magnitude of $(\mathbf{a} + \mathbf{b})$ is $\boxed{4.72}$ .

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