"find a unit vector in the direction of the given unit vector"
vector w = 5i + 5j
apparently i = < 1,0> and j = < 0,1> , and i worked out that the magnitude was 1.
what i'm lost on is that in the example problem it somehow found the direction as well, but in the back of the book, it's looking for only an answer using i and j, but then couldn't i just use the initial equation? ?
for your c:

So what you can do with vectors is make it like a right triangle. In basic terms, you have i=x direction, j=y direction and k=z direction. You have a line that basic goes right 5 and up 5, which is a right triangle with 2 even sides, so it goes up at a 45 degree angle from horizontal.

Now if you got that down pat, a vector is a distance and direction. Direction is the ratio between i-hat, j-hat, and k-hat.

The question asks to find a vector with the same direction, so match the ratios. 5i+5j+0k

examples:
i+j
2i+2j
3i+3j
...

all go in the same direction, 45 degrees from horizontal.

i = <1,0> and j = <0,1> are pretty commonly used to represent the horizontal and vertical unit vector respectively.

vector w = 5i + 5j

has magnitude of sqrt(5^2 + 5^2)  and direction of 45deg.

both can be obtained from making a right-angled triangle with 2 sides at 5.

the prob asks for a unit vector in the same direction as w.

by definition, a unit vector has magnitude of 1, e.g. like i and j.

as w has magnitude of sqrt(5^2 + 5^2) = 5*sqrt(2)

a unit vector in the same direction is w/5*sqrt(2)

= (5i + 5j)/5*sqrt(2)

= 1/sqrt(2) i + 1/sqrt(2) j

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