# Vectors: Problems with Solutions

#### Module or magnitude

$A=(x_1, y_1)$, $B=(x_2, y_2)$
$|\overrightarrow{AB}|=\sqrt{(x_2-x_1)^2+(y_2 - y_1)^2}$

#### Addition and subtraction of vectors

Sum of two vectors - $\vec{S}$ is the result of addition of $\vec{A}$ and $\vec{B}$

Subtraction of vectors

Difference between addition and subtraction of vectors

#### Parallelogram rule for adding vectors

$$|\vec{a}+\vec{b}|=\sqrt{|\vec{a}|^2+|\vec{b}|^2+2\cdot|\vec{a}|\cdot |\vec{b}|\cdot \cos \alpha}$$

#### Angle between vectors

Problem 1
Find the sum of the vectors $\overrightarrow{u}=⟨2,-1⟩$ and $\overrightarrow{v}=⟨3,5⟩$
Problem 2
Given vectors $\overrightarrow{u}=⟨2,-1,3⟩$   $\overrightarrow{v}=⟨3,5,0⟩$    $\overrightarrow{w}=⟨-2,0,3⟩$

$\alpha =2\qquad \beta =-2\qquad \gamma =4$
Find $\alpha \overrightarrow{u}+\beta \overrightarrow{v}+\gamma \overrightarrow{w} =$
Problem 3
Find the direction of the vector $\overrightarrow{v}=⟨3,5⟩$
Problem 4
Find the magnitude of the vector $\alpha \overrightarrow{u}$ if $\alpha =2$ and $\overrightarrow{u}=⟨-2,4,1⟩$
Problem 5
Given two vectors $\overrightarrow{u}=⟨2,3,1⟩$ and $\overrightarrow{v}=⟨-2,5,1⟩.$
Find the magnitude of the vector $\overrightarrow{z}=2\overrightarrow{u}-3\overrightarrow{v}$
Problem 6
Find the magnitude of $\overrightarrow{v}$.

Problem 7
Find the vector sum $\overrightarrow{AB}+\overrightarrow{CA}$, if $A=⟨2,0,-3⟩$, $B=⟨1,1,-2⟩$ and $C=⟨0,3,2⟩$
Problem 8
Find the sum of the vectors $\overrightarrow{u}+\overrightarrow{v}+\overrightarrow{w}$.

Problem 9
Using triangle method graph $\overrightarrow{u}-\overrightarrow{v}$

You do not need to provide a solution. Just open the solution.
Problem 10
Given three vectors $\overrightarrow{u}=⟨2,3,1⟩$, $\overrightarrow{v}=⟨-2,5,1⟩$ and $\overrightarrow{z}=\overrightarrow{u}-2\overrightarrow{v}$. Find the magnitude of the vector $3\overrightarrow{z}$

Problem 11
Find the resultant vector of $\overrightarrow{u}-2\overrightarrow{v}+3\overrightarrow{w}$.

Problem 12
Find the vector $\left( \left\vert \overrightarrow{u}\right\vert +\left\vert \overrightarrow{v}\right\vert-\left\vert \overrightarrow{w}\right\vert \right) \left( \overrightarrow{w}-\overrightarrow{u}\right)$

Problem 13
Find the direction of the vector
$-\left\vert \overrightarrow{u}\right\vert \cdot \overrightarrow{u}$.
$\overrightarrow{u}=⟨-2,3⟩$
Problem 14
Find the vector $\left( \frac{\left\vert \overrightarrow{u}\right\vert +\left\vert \overrightarrow{v}\right\vert }{\left\vert \overrightarrow{w}\right\vert }\right) \left( \overrightarrow{u}-\overrightarrow{v}+\overrightarrow{w}\right)$

Problem 15
The coordinates of the points A, B and C are
$A=(-1,1,2)$, $B=(1,-2,3)$, $C=(-2,1,1)$.
Find the vector sum of $\overrightarrow{u}+\overrightarrow{v}-\overrightarrow{w}$
if $\overrightarrow{u}=\overrightarrow{AB},\overrightarrow{v}=\overrightarrow{CB},\overrightarrow{w}=\overrightarrow{CA}$
Problem 16
Given points A, B and C and constants $\alpha, \beta, \gamma$ such that $A=(-1,2,1)$, $B=(0,2,-3)$, $C=(1,2,-1)$ and $\alpha=-2$, $\beta=2$, $\gamma =3$

$\overrightarrow{u}=\overrightarrow{CB}$
$\overrightarrow{v}=\overrightarrow{AC}$
$\overrightarrow{w}=\overrightarrow{BA}$
Find the vector $\alpha \overrightarrow{u}+\beta \overrightarrow{v}+\gamma \overrightarrow{w}$

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