*Further Vectors

Question 1

A scalar product is equal to a vector (u) multiplied by a vector (v) and it is…
or u•v =…

Answer 1

|u||v|cosθ (with θ being the angle between the vectors when placed tail to tail)

OR

(x1x2) + (y1y2), multiply the x’s multiply the y’s and add them together

Question 2

A scalar product (dot product) is…

Answer 2

A scalar, so direction doesn’t matter since it’s a number (scalar product is |u||v|cosθ)

Question 3

A vector (u) multiplied by itself is...
or u•u = ?

Answer 3

|u|^2 = |u| x |u|

Question 4

When are two vectors (u and v) multiplied equal to 0?

Answer 4

|u||v|cos90 = 0 (the vectors are perpendicular)

or if u = 0

or if v = 0

Question 5

When are two vectors (u and v) that are multiplied parallel and how can they be annotated?

Answer 5

u•v = |u||v|cos0 = |u||v| (the vectors are parallel, like directions)

u•v = |u||v|cos180 = -|u||v| (the vectors are parallel, unlike directions)

or u = λ(v)

Question 6

In a vector projection proj(v)u, which vector is putting their shadow over the other, AND who does all the work (is more present in equation)?

Answer 6

proj(v)u = (v)(u•v)/|v|^2 or (v)(u•v)/(v•v)

v does all the work and u’s ‘shadow’ is on vector v (imagine a sun on top of u, with u’s shadow going onto v)

Question 7

In a vector projection proj(v)u, is it a vector or a scalar and why?

Answer 7

A vector because u•v = |u||v|cosθ = scalar
v•v = |v|^2 = scalar
(v) = vector

(scalar/scalar) x vector = vector

Question 8

In a vector projection proj(v)u, how can you find the scalar projection?

Answer 8

scalar proj(v)u = (u•v)/|v|

it represents the magnitude of proj(v)u can can be found by changing (v)(u•v)/|v|^2 into (v)/|v| x (u•v)/|v| or just (u•v)/|v|

Question 9

u•u is equal to two different expressions of the same scalar product, what are they?

Answer 9

u•u = |u|^2 = |u| x |u|

Question 10

What does the absolute value symbol on a vector also mean?

Answer 10

It is describing the length of the vector

so if v = 5, 30°, then |v| = 5