Length, angle, and right angles

Mathematics · Linear Algebra · ages 19-20

Name ______________________ Date ____________

What is the length of (3, 4)?
Answer: ______________
What is the dot product of (2, 3) and (4, -1)?
Answer: ______________
What is the dot product of (2, 3) and (4, minus 1)?
Answer: ______________
Which vector is orthogonal to (1, 2)?
- (1, 2)
- (2, 1)
- (2,-1)
With orthonormal basis e1 = (3/5, 4/5), e2 = (minus 4/5, 3/5), v = (1, 2) has first coordinate v dot e1. What is it?
Answer: ______________
With orthonormal basis e1 = (3/5, 4/5), e2 = (-4/5, 3/5), v = (1, 2) has first coordinate v·e1. What is it?
Answer: ______________
An orthogonal set of nonzero vectors is always...
- Independent
- Linearly dependent
- A basis of R^2
- Pairwise parallel
The rule <u, v> = u1 v1 minus u2 v2 is an inner product on R squared. True or false?
Circle one: True False
Fred tests <u, v> = u1 v1 + 2 u2 v2 and finds <(0, 1), (0, 1)> = 2. Does that single check prove it is an inner product?
- Yes, positivity is the only axiom
- No, check all axioms
- No, the value 2 disqualifies it
With the same basis and v = (1, 2) as above, what is the second coordinate v dot e2?
Answer: ______________

Answer key

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

5 · sqrt(9 + 16) = sqrt(25) = 5.
5 · 2*4 + 3*(-1) = 8 - 3 = 5.
5 · 2 times 4 + 3 times (minus 1) = 8 minus 3 = 5.
(2,-1) · Only (2,-1) gives a zero dot product: 1 times 2 + 2 times (minus 1) = 0.
2.2 · v dot e1 = 3/5 + 8/5 = 11/5 = 2.2.
2.2 · v·e1 = 3/5 + 8/5 = 11/5 = 2.2.
Independent · Dotting a dependence relation with each member forces every coefficient to zero.
False · False: positivity fails since <(0, 1), (0, 1)> = minus 1.
No, check all axioms · One passing test never certifies a rule; every axiom must hold for all vectors.
0.4 · v dot e2 = minus 4/5 + 6/5 = 2/5 = 0.4.