Question

Magnitude of Vectors

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

$\mathbf{u}=⟨10,-1⟩,\mathbf{v}=⟨-2,-2⟩$

Short answer

The required values are-

$\begin{array}{l}\left|u\right|=\sqrt{101}\\ \left|v\right|=\sqrt{8}\\ \left|2u\right|=2\sqrt{101}\\ \left|\frac{1}{2}v\right|=\sqrt{2}\\ |u+v|=\sqrt{73}\\ |u-v|=\sqrt{145}\\ \left|u\right|-\left|v\right|=\sqrt{101}-\sqrt{8}\end{array}$

Step-by-step solution

Step 1. Given information

Two points are given as $\mathbf{u}=⟨10,-1⟩,\mathbf{v}=⟨-2,-2⟩$

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,

$\begin{array}{l}\left|u\right|=\sqrt{{10}^2+{(-1)}^2}\\ =\sqrt{100+1}\\ =\sqrt{101}\\ \left|v\right|=\sqrt{{(-2)}^2+{(-2)}^2}\\ =\sqrt{4+4}\\ =\sqrt{8}\end{array}$

Also in,

$\begin{array}{l}\left|2u\right|=2\left|u\right|\\ =2\sqrt{101}\\ \left|\frac{1}{2}v\right|=\frac{1}{2}\left|v\right|\\ =\frac{1}{2}\sqrt{8}\\ =\frac{2\sqrt{2}}{2}\\ =\sqrt{2}\end{array}$

And,

$\begin{array}{l}|u+v|=\sqrt{{(10-2)}^2+{(-1-2)}^2}\\ =\sqrt{8^2+3^2}\\ =\sqrt{64+9}\\ =\sqrt{73}\\ |u-v|=\sqrt{{(10-(-2\left)\right)}^2+{(-1-(-2\left)\right)}^2}\\ =\sqrt{{12}^2+1^2}\\ =\sqrt{144+1}\\ =\sqrt{145}\\ \left|u\right|-\left|v\right|=\sqrt{101}-\sqrt{8}\end{array}$