One number that knows the angle

Mathematics · Geometry · ages 19-20

Name ______________________ Date ____________

What is the angle between (1, 0) and (0, 1)?
- 0 degrees
- 90 degrees
- 45 degrees
- 180 degrees
Compute the dot product (1, 2) dot (3, 4).
Answer: ______________
Compute the dot product (1, 2) dot (3, 4).
Answer: ______________
Compute the dot product (2, 3) dot (minus 1, 4).
Answer: ______________
What is the scalar component of (1, 2) along the direction (3, 4)?
Answer: ______________
Which pair of vectors is perpendicular?
- (1, 2) and (-2, 1)
- (1, 2) and (2, 1)
- (1, 1) and (2, 3)
- (3, 0) and (1, 1)
Let u = (2, 1) and v = (-1, 2). Sam says u and v are perpendicular. Is Sam right?
Circle one: True False
What is the scalar component of (1, 2) along the direction (3, 4)?
Answer: ______________
What is the projection vector of (4, 0) onto the direction (1, 1)?
- (2, 2)
- (4, 4)
- (2, 0)
Mia says two nonzero vectors with dot product zero must be perpendicular. Is Mia right?
Circle one: True False

Answer key

For grown-ups. Fold this page away before handing over the rest.

90 degrees · The dot product is 0, and both vectors are unit length, so cosine is 0, which means 90 degrees.
11 · Multiply matching components and add: 3 + 8 = 11.
11 · Multiply matching parts and add: 3 + 8 = 11.
10 · 2 times minus 1 plus 3 times 4 is minus 2 plus 12 = 10.
2.2 · Normalize (3, 4) to (3/5, 4/5), then dot with (1, 2): 3/5 + 8/5 = 11/5 = 2.2.
(1, 2) and (-2, 1) · Only the first pair dots to zero: minus 2 plus 2 = 0.
True · Sam is right: the dot product is minus 2 plus 2, which is 0.
2.2 · Normalize (3, 4) to (3/5, 4/5), then dot: 3/5 + 8/5 = 11/5 = 2.2.
(2, 2) · Dot with (1, 1) to get 4, divide by squared length 2, and scale (1, 1) by 2.
True · Zero dot with nonzero lengths gives cosine 0, which is exactly 90 degrees.