Vector Basics
What is a quantity that has both magnitude and direction?
Vector
What geometric method can be used to add two vectors by placing the tail of one vector at the head of the other?
The head-to-tail method.
A vector has a magnitude of 10 and makes an angle of 30∘ with the positive x-axis. Find its horizontal component.
10 Cos30=5(3)^1/2
What happens to the direction and magnitude of a vector when it is multiplied by −1?
Its direction reverses, while its magnitude stays the same.
In triangle ABC,
AB=a
and
AC=b.
Express BC in terms of a and b.
b-a
What is the magnitude of a vector represented by a 3-unit horizontal displacement and a 4-unit vertical displacement?
5
If ∣a∣=5 and ∣b∣=3, what is the maximum possible magnitude of a+b?
8
A vector has a magnitude of 8 and makes an angle of 60∘ with the positive x-axis. Find its vertical component.
8*sin60=4(3)^1/2
If two non-zero vectors are parallel and point in the same direction, how can one vector be written in terms of the other?
b=ka, k>0
A vector
a=[k,4]
has magnitude 5. Find all possible values of k.
k=+/-3
Two vectors are equal when they have the same ______ and ______.
Magnitude and direction
If ∣a∣=5 and ∣b∣=3, what is the minimum possible magnitude of a+b?
2
Find the magnitude of the vector
v=6i+8j.
∣v∣=(6^2+8^2)^1/2=10
Two vectors have magnitudes 5 and 8, and the angle between them is 60∘. Find the magnitude of their sum.
∣a+b∣=[5^2+8^2+2(5)(8)cos60∘]^1/2 =
[25+64+40 *129]^1/2≈11.36
Find k if the vectors
a=[k,2]
and
b=[3,−6]
are perpendicular.
a⋅b=0
3k−12=0
k=4
What is the difference between a scalar and a vector?
A scalar has magnitude only; a vector has magnitude and direction.
If a+b=0, what can you conclude about vectors a and b?
They have the same magnitude but opposite directions.
Find the direction angle of the vector
v=3i+3j.
θ=tan−1(3/3)=45∘
Given ∣a∣=7, ∣b∣=9, and the angle between them is 60∘, find ∣a−b∣.
671/2
Vectors a=[2,3] and b=[x,4]
have an angle of 60∘ between them. Find x.
a⋅b=∣a∣∣b∣cos60∘
2x+12=(13^1/2 )(x^2+16)(1/2))
Squaring and solving gives:
x=2
If vector a has a magnitude of 7, what is the magnitude of −3a?
21
ector a has a magnitude of 10 and points east. Vector b has a magnitude of 6 and points west. Find the magnitude of a+b and direction.
4, East direction
A vector has components (−5,12). Find its magnitude and its direction angle measured counterclockwise from the positive x-axis.
Magnitude:
∣v∣=[(−5)^2+12^2]^1/2=13
Since the vector is in Quadrant II,
θ=180∘−tan−1(5/12) θ≈112.6∘
Vectors a and b have magnitudes 10 and 6. If their resultant has magnitude 14, determine the angle between them.
14^2=10^2+6^2+2(10)(6)cosθ
196=136+120cosθ
cosθ=1/2
60∘
A ship travels with velocity
v=[20,15] km/h
relative to the water. The current has velocity
c=[−8,6] km/h.
a) Find the ship's actual velocity relative to the ground.
b) Find its speed.
c) Find its direction measured counterclockwise from the positive x-axis.
a. vg=v+c =[20,15]+[−8,6]
vg=[12,21] km/h
b. ∣vg∣=[12^2+21^2 ]^1/2=365≈24.19 km/h
c. θ=tan−1(12/21)
θ≈60.3∘