Dot Product
If a = <a1, a2, a3> and b = <b1, b2, b3>, then the dot product of a and b is the number a · b given by
a · b = a1b1 + a2b2+ a3b3
What are the angles α, β, γ (in the interval [0,π]) that a makes with the positive x-, y-, and z-axes?
The direction angles of the nonzero vector a
Let u and v be nonzero vectors. The vector projection of u onto v is:
projvu = [(u · v)/(|v|2)] v
By the Direction Cosine Identity, cos2α + cos2β + cos2γ = ___
1
What is the formula for work?
W = | F | | D | cos θ = F · D
Let c be a scalar.
c (a · b) = (ca) · b = a · (cb)
What is the property?
Associative Property
Can a vector have direction angles α = 45°, β = 60°, and γ = 30°?
No
Find the scalar projection of b = <1,0,1> onto a = <1, 1, -1>.
0
Find the angle between a = <1,1> and b = <0,5>.
45°
Given F = <4,2>, and d = <2,4>, calculate the work between them.
16
Given: a = i - 2j + 3k and b = 5i + 9k
Find a ⋅ b
32
Calculate the direction cosines for vector v = 2i + 4j - 6k.
cos α ≈ 2/√56
cos β ≈ 4/√56
cos γ ≈ -6/√56
Find the scalar and vector projections of a = <3, 4> and b = <5, 2>.
23/√29, <115/29, 46/29>
Find the angle between u = <4,3> and v = <3,5>. Round to the nearest tenth.
~22.2°
Given F = <6,7>, d = (6,7) calculate the work between them.
85
Use the vectors u = <2,2>, v = <-3,4>, and w = <1,-2> to find (v · u) - (w · v). State whether the result is a vector or a scalar.
13, scalar
Given a vector v with magnitude ||v|| =10, and two of its direction angles are α = 45° and β = 60°, find the possible values for the direction angle γ?
60° or 120°
Find the scalar and vector projections of ⟨1, 2, 3⟩ onto ⟨1, 2, 0⟩.
√5, ⟨1, 2, 0⟩
Use the dot product to find the magnitude of u = 12i - 16j
20
Given F = <1,3,2> and d = <2,3,2>, calculate the work between them.
15
The vector u = <1650, 3200> gives the number of two types of baking pans produced by a company. The vector v = <15.25, 10.50> gives the prices (in dollars) of the two types of pans, respectively. Find the dot product and interpret the result in the context of the problem.
In the context of this problem, the dot product $58,762.50 represents the total revenue (in dollars) generated from selling all units of both types of baking pans.
Find the direction angles α, β, and γ for the vector a = 2i + 3j - 6k.
α ≈ 73.4°
β ≈ 64.6°
γ ≈ 148.9°
Find the scalar and vector projections of ⟨1, 1, 1⟩ onto ⟨3, 2, 1⟩.
3√14/7, ⟨9/7, 6/7, 3/7⟩
For what value(s) of k are the vectors u = <2, -4, 1> and v = <-6, k, -3> perpendicular?
k = -3.75
Given F = <2,4,2> and d = <2,4,2>, calculate half the work between them.
12