Vector addition subtraction and magnitude with coordinates

Key concept: This problem combines basic vector arithmetic with magnitude and unit-vector notation, so each part should be handled one operation at a time.

Step by step

Vector addition and subtraction

For vectors written in component form, add or subtract the $x$-components and $y$-components separately.

Given

$

$\begin{bmatrix}5\\2\end{bmatrix}+\begin{bmatrix}3\\-2\end{bmatrix}=\begin{bmatrix}8\\0\end{bmatrix}$

$

and

$

$\begin{bmatrix}5\\2\end{bmatrix}-\begin{bmatrix}3\\-2\end{bmatrix}=\begin{bmatrix}2\\4\end{bmatrix}$

$

These are direct component-wise operations.

Dot product and magnitude

For the vector

$

$P=\begin{bmatrix}2\\-2\end{bmatrix}\cdot\begin{bmatrix}3\\-7\end{bmatrix}$

$

compute the dot product first:

$

$P=(2)(3)+(-2)(-7)=6+14=20$

$

If the work shown in the prompt is intended to represent a vector with components $6$ and $-14$, then its magnitude is found using

$

$|P|=\sqrt{6^2+(-14)^2}=\sqrt{36+196}=\sqrt{232}$

$

That magnitude is then used to create a unit vector.

Unit vector form

A unit vector has length 1, so divide the vector by its magnitude:

$

$\hat P_C=\frac{1}{\sqrt{232}}\begin{bmatrix}6\\-14\end{bmatrix}$

$

Likewise, for the vector $a=\begin{bmatrix}3\\2\end{bmatrix}$, its magnitude is

$

$|a|=\sqrt{3^2+2^2}=\sqrt{13}$

$

so the unit vector is

$

$\hat a=\frac{1}{\sqrt{13}}\begin{bmatrix}3\\2\end{bmatrix}$

$

Common method reminders

When working with vectors, keep three ideas separate: addition/subtraction changes components, the dot product gives a scalar, and magnitude measures length. Unit vectors always keep the same direction and reduce the vector to length 1.

Final answers from the displayed work

$

$\begin{bmatrix}5\\2\end{bmatrix}+\begin{bmatrix}3\\-2\end{bmatrix}=\begin{bmatrix}8\\0\end{bmatrix},\quad \begin{bmatrix}5\\2\end{bmatrix}-\begin{bmatrix}3\\-2\end{bmatrix}=\begin{bmatrix}2\\4\end{bmatrix}$

$

$

$|a|=\sqrt{13},\quad \hat a=\frac{1}{\sqrt{13}}\begin{bmatrix}3\\2\end{bmatrix}$

$

Pitfall alert

A very common mistake is to add magnitudes instead of components. Vectors do not combine by length alone; each coordinate must be handled separately. Another frequent error is forgetting that a unit vector is the original vector divided by its own magnitude, not by the magnitude of a different vector. Also, in a dot product, the result is a single number, not a vector, so do not keep it in bracket form unless the question explicitly asks for a vector quantity.

Try different conditions

If the vector were changed to $a=\begin{bmatrix}-3\\2\end{bmatrix}$, the magnitude would still be $\sqrt{13}$, but the unit vector would become $\frac{1}{\sqrt{13}}\begin{bmatrix}-3\\2\end{bmatrix}$. If the operation were $\begin{bmatrix}5\\2\end{bmatrix}+\begin{bmatrix}3\\2\end{bmatrix}$ instead, the result would be $\begin{bmatrix}8\\4\end{bmatrix}$. The method stays the same: work component by component, then simplify the result.

Further reading

vector components, unit vector, dot product