Question

Magnitude of Vectors

Find $\left|\mathbf{u}\right|,\left|\mathbf{v}\right|,\left|2\mathbf{u}\right|,\left|\frac{1}{2}\mathbf{v}\right|,|\mathbf{u}+\mathbf{v}|,|\mathbf{u}-\mathbf{v}|\&\left|\mathbf{u}\right|-\left|\mathbf{v}\right|$

$\mathbf{u}=⟨-6,6⟩,\mathbf{v}=⟨-2,-1⟩$

Short answer

The required values are-

$\begin{array}{l}\left|u\right|=\sqrt{72}\\ \left|v\right|=\sqrt{5}\\ \left|2u\right|=2\sqrt{72}\\ \left|\frac{1}{2}v\right|=\frac{\sqrt{5}}{2}\\ |u+v|=\sqrt{89}\\ |u-v|=\sqrt{65}\\ \left|u\right|-\left|v\right|=\sqrt{72}-\sqrt{5}\end{array}$

Step-by-step solution

Step 1. Given information

Two points are given as $\mathbf{u}=⟨-6,6⟩,\mathbf{v}=⟨-2,-1⟩$

Step 2. Concept used

Modulus represents the magnitude of vector which can be written as-

$\left|v\right|=\sqrt{a_1^2+a_2^2}$ for a point $u=a_{1}i+a_{2}j$.

We can also add vectors as-

$u+v=\left(a_1+b_1\right)i+\left(a_2+b_2\right)j$

Use the vector formulas to get the results.

Step 3. Calculation

Substitute the values of and in the required terms,

$\begin{array}{l}\left|u\right|=\sqrt{{(-6)}^2+6^2}\\ =\sqrt{36+36}\\ =\sqrt{72}\\ \left|v\right|=\sqrt{{(-2)}^2+{(-1)}^2}\\ =\sqrt{4+1}\\ =\sqrt{5}\end{array}$

Also in,

$\begin{array}{l}\left|2u\right|=2\left|u\right|\\ =2\sqrt{72}\\ \left|\frac{1}{2}v\right|=\frac{1}{2}\left|v\right|\\ =\frac{1}{2}\sqrt{5}\\ =\frac{\sqrt{5}}{2}\end{array}$

And,

$\begin{array}{l}|u+v|=\sqrt{{(-6-2)}^2+{(6-1)}^2}\\ =\sqrt{64+25}\\ =\sqrt{89}\\ |u-v|=\sqrt{{(-6-(-2\left)\right)}^2+{(6-(-1\left)\right)}^2}\\ =\sqrt{16+49}\\ =\sqrt{65}\\ \left|u\right|-\left|v\right|=\sqrt{72}-\sqrt{5}\end{array}$