Quiz #1
MATH 151 Section
Name:
September 5, 2002
1. Calculate the vector 2 h3, −7i − 5 h13, 5i and give your answer in terms of the standard basis
vectors i and j.
2 h3, −7i − 5 h13, 5i = h6, −14i − h65, 25i = h−59, −39i
Answer: −59i − 39j
2. Find a unit vector that points in the same direction as the vector starting at the point (5, 13)
and going to (17, 11).
√
p
→
−
→
−
The vector is b = h17 − 5, 11 − 13i = h12, −2i and its length is | b | = 122 + (−2)2 = 148
1
To make a vector into a unit vector, multiply it by −
→.
|b|
Answer:
12
−2
√
,√
148
148
≈ h.9864, −.1644i
−
−
3. If →
a = −3i + 5j, what angle does →
a make with the positive x axis?
Notice we want the angle θ that is indicated in the picture.
α
θ
α = arctan 53 and θ = 180o − α
Answer: θ = 120.9638o or 2.111 radians
4. A bird is flying northeast (i.e. with a bearing of N45o E) at a speed of 8mph. There is a slight
breeze blowing due north at a speed of 1 mph. What vector would represent the resulting
path of the bird? What is the ground speed (resulting speed) of the bird?
→
−
The bird’s vector is b = h8 cos(45), 8 sin(45)i ≈ h5.6569, 5.6569i
→
The wind vector is −
w = h0, 1i
→ →
−
→
Answer: The resulting vector √
is −
r = b +−
w = h5.6569, 6.6569i
→
−
The resulting speed is | r | = 8.735mph.