Question

4. For v = -2i – 7j, find the unit vector in the direction of y, and write your answer as a linear combination of the standard unit vectors i and j

5. Find u•v if u = i + 2j and v = -i + 5j.​

Answer 1

4. If v = -2i - 7j, then its magnitude is

||v|| = √((-2)² + (-7)²) = √53

and the unit vector in the direction of v is

`(v)/(\|v\|) = \boxed{-\frac2{√(53)} \, \vec\imath - \frac7{√(53)} \, \vec\jmath}`

5. Given u = i + 2j and v = -i + 5j, we have

u • v = (i + 2j) • (-i + 5j) = -(i • i) - 2 (j • i) + 5 (i • j) + 10 (j • j)

Since i and j are orthogonal unit vectors, we have i • i = j • j = 1 and i • j = 0, so the product reduces to

u • v = -1 - 0 + 0 + 10 = 9