When are these vectors parallel/perpendicular/the same length?

When are these vectors parallel/perpendicular/the same length?

Question

Given the vectors $\mathbf{U}=\begin{pmatrix}7 \\ 6\end{pmatrix}$ and $\mathbf{V}=\begin{pmatrix}2 \\ x\end{pmatrix}$, find the value or values of $x$ for which

1. $\mathbf{U}$ is parallel to $\mathbf{V}$,

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1. $\mathbf{U}$ is perpendicular to $\mathbf{V}$,

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1. $\mathbf{U}$ and $\mathbf{V}$ have the same magnitude.

Solution

Two vectors $\mathbf{U}$ and $\mathbf{V}$ are parallel if and only if $\mathbf{U}=k\mathbf{V}$ for some scalar $k$.

This happens in our case if and only if $7 = k \times 2$, and $6 = k \times x$. This means $k = 3.5$, and so $x = \dfrac{12}{7} = 1.71 \quad (3$sf).

Given the vectors $\mathbf{U}=\begin{pmatrix}7 \\ 6\end{pmatrix}$ and $\mathbf{V}=\begin{pmatrix}2 \\ x\end{pmatrix}$, find the value or values of $x$ for which

1. $\mathbf{U}$ is perpendicular to $\mathbf{V}$,

Two vectors in two dimensions are perpendicular if and only if their gradients multiply to $-1$.

This happens here if and only if

$$\dfrac{6}{7} \times \dfrac{x}{2} = -1 \implies x = -\dfrac{7}{3}= -2.33 \quad (3\text{sf}).$$

Given the vectors $\mathbf{U}=\begin{pmatrix}7 \\ 6\end{pmatrix}$ and $\mathbf{V}=\begin{pmatrix}2 \\ x\end{pmatrix}$, find the value or values of $x$ for which

1. $\mathbf{U}$ and $\mathbf{V}$ have the same magnitude.

We know that the magnitude of $\begin{pmatrix}a \\ b\end{pmatrix}$ is $\sqrt{a^2+b^2}$.

So we need in our case to have

$\sqrt{7^2+6^2}=\sqrt{x^2+2^2}$

, which gives us that

$$\begin{align*} x^2 &= 7^2+6^2-2^2 \\ x &= \pm 9. \end{align*}$$

We can check our answers for all three parts using the GeoGebra applet below.

[interactive embed (geogebra-iframe): //www.geogebratube.org/material/iframe/id/KzQhNUqH/width/600/height/600/border/888888/rc/false/ai/false/sdz/false/smb/false/stb/false/stbh/false/ld/false/sri/true/asb/false/at/html5] — not part of the text import; see the source page.

Download GeoGebra file