Use the given pair of vectors, u = −i − 2j and v = 4i − 5j to calculate the following

Question

Use the given pair of vectors, u = −i − 2j and v = 4i − 5j to calculate the following

1. Are u and v orthogonal? Explain how the dot product can help you determine this.
2. proj_v(u), and classify wether it is scalar or vector.
3. Calculate the angle between u and v. Give the exact angle first (simplify inside of square roots; multiplying square roots is optional), then use a calculator to approximate the angle in degrees rounded to 1 decimal place.

Answer 1

they are orthogonal (perpendicular) if the dot product= 0

dot product of u and v is not zero. they're not orthogonal

angle between them must = 90 degrees, to be orthogonal. the angle > 90

the projection is a vector, not a scalar, vectors have magnitude and direction

the angle between u and v is the arccos(dot product divided by product of absolute values of u and v)

u= <-i,-2j>, v=<4i,-5j>

angle= arccos(uv/|u||v|)=arccos((-4+10)/sqr(5(41))=arccos(6/sqr205)

=about 65.2 degrees

as a rough check, sketch the vectors and visually see if about 65 looks in the right range

or just look at the sketch and realize it's no close to 90 degrees

or use the arcsine formula with cross products