Chapter 1: Vectors, Lines and Planes
Section 1.3: Dot Product
The dot product of two vectors is a scalar. Table 1.3.1 summarizes the definition, essential properties, and useful consequences of this product. In the table, a vector A has components ak,k=1,…,n, where n can be 2 or 3 or even an integer greater than 3. Likewise for a vector B.
Because the dot product yields a scalar, some texts call it the scalar product of two vectors.
Norm of A
Points A and B
A−B or B−A, where A and B are position vectors to points A and B
A·B=ABcos(θ), where θ is the angle between A and B
Angle between Vectors
Orthogonality of A and B
A·B=0⇒A ⊥ B, provided neither A nor B is the zero vector
Direction Cosines for A
cosα=A·iA = a1A
cosβ=A·jA = a2A
cosγ=A·kA = a3A
Scalar Projection of B on A
BA = B·AA
Vector Projection of B on A
BA=B·AAAA = B·AA·AA
Component of B orthogonal to A
Product Rule of Differentiation
Table 1.3.1 The dot product, its definition, properties, and consequences
The angles α,β,γ are called the direction angles.
A common multiple of the direction cosines generates a set of direction numbers.
This Study Guide will use the notation BA for the vector projection of B upon A.
This Study Guide will use the notation B⊥A for the component of B orthogonal to A.
Some texts will call the dot product the "scalar product."
Some texts will write A2 for A·A.
Some texts will write A for A, the magnitude of A.
The product rule for differentiation of a dot product is consistent with that for differentiation of a product of scalars: "first times the derivative of the second plus the second times the derivative of the first", with "times" replaced by "dot".
Implementing the Dot Product in Maple
A DotProduct command appears in four of the five relevant Maple packages listed in Table 1.3.2. Surprisingly, it does not exist in the Student LinearAlgebra package where it is only implemented with the period or with the heavier dot found in the Common Symbols palette. However at top-level, and in all five listed packages, the Context Panel provides an interactive dot product option.
At top-level and in the LinearAlgebra package, Maple takes the dot product over the complex numbers, so the first vector is conjugated. This only affects vectors whose components are variables.
Table 1.3.2 summarizes the location and implementation of the dot product in Maple.
Period or · from Common Symbols palette
Table 1.3.2 Implementing the dot product in Maple
If A=3 i+7 j and B=−4 i+5 j,
Obtain θ, the angle between A and B
Verify that A2=A·A
Obtain the scalar projection of B on A
Obtain the vector projection of B on A
Obtain the component of B orthogonal to A
If A=3 i+2 j+7 k and B=4 i−5 j+6 k,
For A, obtain direction cosines, angles, and numbers.
Using the law of cosines, verify the equivalence of A·B and ABcos(θ)
In the xy-plane, obtain all vectors that are orthogonal to A=3 i+2 j.
If A=3 i+2 j+5 k and B=−4 i+5 j+λ k, find all values of λ that make B orthogonal to A.
Use vector methods to find the distance between the points A:4,5 and B:−3,2.
Use vector methods to find the distance between the points A:4,5,7 and B:−3,2,3.
Suppose the components of the planar vectors A and B are functions of t, and the derivative of such vectors is defined to be the vector of componentwise derivatives. If the prime denotes differentiation with respect to t, show that
If A is a unit vector whose components are functions of t, show that A and A′ are necessarily orthogonal.
Verify this for A=cost i+sint j.
Verify this for A, the normalization of V=t i+t2 j.
Find a vector of unit length that is orthogonal to the vectors A=i+j+k and B=2 i+3 j−k.
Show that A·A=0 is a necessary and sufficient condition for A to be the zero vector.
Use vector methods to prove that an angle inscribed in a semicircle is necessarily a right angle.
Do the terminal points of the position vectors to points A:2,−3,7, B:1,−1,10,
and C:3,−5,4 lie on a straight line?
If A and B are unit vectors with θ the angle between them, show that sinθ/2=12A−B.
If A=ax i+bx j and B=ux i+vx j, verify the product rule for differentiating the dot product A·B.
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