Evaluate the expression:
v ⋅ w
Given the vectors:
r = <5, -5, -2>; v = <2, -8, -8>; w = <-2, 6, -5>
Vectors are quite simple.
I'm assume thats $\displaystyle V - W $. Just do component wise subtraction.
For example
$\displaystyle \begin{bmatrix} 5 \\ 6 \end{bmatrix} + \begin{bmatrix} 7 \\ 8 \end{bmatrix} = \begin{bmatrix} 5 + 7 \\ 6 + 8 \end{bmatrix} $
Likewise for subtraction
$\displaystyle \begin{bmatrix} 5 \\ 6 \end{bmatrix} - \begin{bmatrix} 7 \\ 8 \end{bmatrix} = \begin{bmatrix} 5 - 7 \\ 6 - 8 \end{bmatrix} $
Have you ever taken a course involving vectors? If not where did you get this problem? If so, you should have learned that there are a number of operations involving vectors:
1) Addition: <a, b, c>+ <d, e, f>= <a+ d, b+ e, c+ f>.
2) Subtraction: <a, b, c>- <d, e, f>= <a- d, b- e, c- f>.
3) Scalar multiplication: d<a, b, c>= <ad, bd, cd> where d is a "scalar" (number).
4) Dot product or inner product: <a, b, c> . <d, e, f>= ad+ be+ cf. Notice the result is a number, not a vector.
What you wrote is very difficult to read. jakncoke assumed the symbol between the two vectors was a "-" and that you meant to subtract the two vectors:
v- w= <2, -8, -8>- <-2, 6, -5>= <2- (-2), -8- 6, -8- (-5)>= <4, -14, -3>.
It looks to me like "." or dot product: <2, -8, -8>.<-2, 6, -5>= 2(-2)+ (-8)(6)+ (-8)(-5)= -4- 14+ 40= -18+ 40= 22.
A vector is basically an arrow pointing the direction and how far youd like to go in that direction. Lets take a 2 dimensional vector, or a vector with 2 components. It basically tells you to go a certain amount of units on the x-axis and then go up a certain amount of units in the y axis
It has the form
$\displaystyle \begin{bmatrix} x \\ y \end{bmatrix} $ where x and y are just numbers like 2,3,4,5.5 etc...
If you want to think visually, this vector tells us to move x units on the X-axis and Y-units in the y axis.
The picture below illustrates graphically what a vector does.
For this example i use the vector $\displaystyle \begin{bmatrix} 2 \\ 2 \end{bmatrix} $
Notice that a vector just tells how where and how far to go from any given point, thats why both the vector (Arrows) colored in black are the same vector $\displaystyle \begin{bmatrix} 2 \\ 2 \end{bmatrix} $
So vector addition, lets take an example, $\displaystyle \begin{bmatrix} 2 \\ 2 \end{bmatrix} + \begin{bmatrix} 3 \\ 4 \end{bmatrix} = \begin{bmatrix} 2 + 3 \\ 2 + 4\end{bmatrix} $ so we have two vectors, one says go 2 units on the x axis and 2 units on the y axis, the second one says go 3 units in the x-axis and 4 units in the y-axis. Adding these two together is like saying, go (2 + 3) or 5 units in the x-axis and then go (2+4) or 6 units in the y-axis.
subtracting vectors is like adding the negative, so $\displaystyle \begin{bmatrix} 2 \\ 2 \end{bmatrix} - \begin{bmatrix} 3 \\ 4 \end{bmatrix} = \begin{bmatrix} 2 \\ 2 \end{bmatrix} + \begin{bmatrix} -3 \\ -4 \end{bmatrix} = \begin{bmatrix} 2-3 \\ 2-4 \end{bmatrix}$ So basically go (2-3) or -1 units on the x-axis and (2-4) or -2 units on the y-axis.
@jakncoke, I don't normally disagree with this sort of reply. But this time your reply is wrongly over-the-top.
Actually vector really does not have a good mathematical definition.
That is, in mathematics we define terms of set theoretic terms.
BUT we know that a vector is an equivalence class of objects that have the same length & direction .
Now it is possible to define each of those mathematically and then put them together.
So the concept of vectors is so much more than your reply seems to understand.