# 10.8 Vectors  (Page 3/22)

 Page 3 / 22

To find u + v , we first draw the vector u , and from the terminal end of u , we drawn the vector v . In other words, we have the initial point of v meet the terminal end of u . This position corresponds to the notion that we move along the first vector and then, from its terminal point, we move along the second vector. The sum u + v is the resultant vector because it results from addition or subtraction of two vectors. The resultant vector travels directly from the beginning of u to the end of v in a straight path, as shown in [link] .

Vector subtraction is similar to vector addition. To find u v , view it as u + (− v ). Adding − v is reversing direction of v and adding it to the end of u . The new vector begins at the start of u and stops at the end point of − v . See [link] for a visual that compares vector addition and vector subtraction using parallelograms .

Given $u$ $=⟨3,-2⟩$ and $v$ $=⟨-1,4⟩,$ find two new vectors u + v , and u v .

To find the sum of two vectors, we add the components. Thus,

$\begin{array}{l}u+v=⟨3,-2⟩+⟨-1,4⟩\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=⟨3+\left(-1\right),-2+4⟩\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=⟨2,2⟩\hfill \end{array}$

To find the difference of two vectors, add the negative components of $\text{\hspace{0.17em}}v\text{\hspace{0.17em}}$ to $\text{\hspace{0.17em}}u.\text{\hspace{0.17em}}$ Thus,

$\begin{array}{l}u+\left(-v\right)=⟨3,-2⟩+⟨1,-4⟩\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=⟨3+1,-2+\left(-4\right)⟩\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=⟨4,-6⟩\hfill \end{array}$

## Multiplying by a scalar

While adding and subtracting vectors gives us a new vector with a different magnitude and direction, the process of multiplying a vector by a scalar    , a constant, changes only the magnitude of the vector or the length of the line. Scalar multiplication has no effect on the direction unless the scalar is negative, in which case the direction of the resulting vector is opposite the direction of the original vector.

## Scalar multiplication

Scalar multiplication involves the product of a vector and a scalar. Each component of the vector is multiplied by the scalar. Thus, to multiply $v$ $=⟨a,b⟩$ by $k$ , we have

$kv=⟨ka,kb⟩$

Only the magnitude changes, unless $\text{\hspace{0.17em}}k\text{\hspace{0.17em}}$ is negative, and then the vector reverses direction.

## Performing scalar multiplication

Given vector $\text{\hspace{0.17em}}v$ $=⟨3,1⟩,\text{\hspace{0.17em}}$ find 3 v , $\frac{1}{2}$ $v,\text{\hspace{0.17em}}$ and − v .

See [link] for a geometric interpretation. If $\text{\hspace{0.17em}}v$ $=⟨3,1⟩,$ then

$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}3v=⟨3\cdot 3,3\cdot 1⟩\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=⟨9,3⟩\hfill \\ \text{\hspace{0.17em}}\frac{1}{2}v=⟨\frac{1}{2}\cdot 3,\frac{1}{2}\cdot 1⟩\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=⟨\frac{3}{2},\frac{1}{2}⟩\hfill \\ -v=⟨-3,-1⟩\hfill \end{array}$

Find the scalar multiple 3 $u$ given $u$ $=⟨5,4⟩.$

$3u=⟨15,12⟩$

## Using vector addition and scalar multiplication to find a new vector

Given $u$ $=⟨3,-2⟩$ and $v$ $=⟨-1,4⟩,$ find a new vector w = 3 u + 2 v .

First, we must multiply each vector by the scalar.

$\begin{array}{l}3u=3⟨3,-2⟩\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=⟨9,-6⟩\hfill \\ 2v=2⟨-1,4⟩\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=⟨-2,8⟩\hfill \end{array}$

$\begin{array}{l}w=3u+2v\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=⟨9,-6⟩+⟨-2,8⟩\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=⟨9-2,-6+8⟩\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=⟨7,2⟩\hfill \end{array}$

So, $w$ $=⟨7,2⟩.$

## Finding component form

In some applications involving vectors, it is helpful for us to be able to break a vector down into its components. Vectors are comprised of two components: the horizontal component is the $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ direction, and the vertical component is the $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ direction. For example, we can see in the graph in [link] that the position vector $⟨2,3⟩$ comes from adding the vectors v 1 and v 2 . We have v 1 with initial point $\text{\hspace{0.17em}}\left(0,0\right)\text{\hspace{0.17em}}$ and terminal point $\text{\hspace{0.17em}}\left(2,0\right).\text{\hspace{0.17em}}$

stock therom F=(x2+y2) i-2xy J jaha x=a y=o y=b
root under 3-root under 2 by 5 y square
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
cosA\1+sinA=secA-tanA
why two x + seven is equal to nineteen.
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
SAIDI
what is the best practice that will address the issue on this topic? anyone who can help me. i'm working on my action research.
simplify each radical by removing as many factors as possible (a) √75
how is infinity bidder from undefined?
what is the value of x in 4x-2+3
give the complete question
Shanky
4x=3-2 4x=1 x=1+4 x=5 5x
Olaiya
hi can you give another equation I'd like to solve it
Daniel
what is the value of x in 4x-2+3
Olaiya
if 4x-2+3 = 0 then 4x = 2-3 4x = -1 x = -(1÷4) is the answer.
Jacob
4x-2+3 4x=-3+2 4×=-1 4×/4=-1/4
LUTHO
then x=-1/4
LUTHO
4x-2+3 4x=-3+2 4x=-1 4x÷4=-1÷4 x=-1÷4
LUTHO
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was  1350  bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after  3  hours?
v=lbh calculate the volume if i.l=5cm, b=2cm ,h=3cm
Need help with math
Peya
can you help me on this topic of Geometry if l help you
litshani
( cosec Q _ cot Q ) whole spuare = 1_cosQ / 1+cosQ
A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So, the length of the guy wire can be found by evaluating √(90000+160000). What is the length of the guy wire?
the indicated sum of a sequence is known as
how do I attempted a trig number as a starter