Chapter 1: Vectors, Lines and Planes
Section 1.5: Applications of Vector Products
Table 1.5.1 lists formulae that combine both dot and cross products.
Triple Scalar (Box) Product
Area of parallelogram with edges A and B
Area of triangle with edges A and B
Volume of parallelepiped with edges A, B, and C
Distance from point P to line through points Q and R.
A is the vector from Q to R; B, the vector from Q to P
Distance from point P to plane through points Q, R, and S.
A is the vector from Q to R; B, from Q to S; and C from Q to P
Vectors V1,V2,V3 and U1,U2,U3 are reciprocal sets of vectors if Ui·Vj=δij=1i=j0i≠j.
V1=U2×U3λ, V2=U3×U1λ, V3=U1×U2λ
Torque τ exerted about O by a force F acting at the head of r, a vector of length r with tail at O
Table 1.5.1 Formulas involving dot and cross products
If A=3 i−2 j+4 j, B=2 i+5 j−4 k, and C=5 i+7 j+6 k,
Compute ABC, the Triple Scalar (or Box) Product A·B×C.
Verify the identity A·B×C=A×B·C for the Triple Scalar Product.
For the vectors A, B, and C of Example 1.5.1, and D=4 i+3 j−2 k,
Verify the identity A×B×C×D=ACDB−BCDA.
Verify the identity A×B×C×D=ABDC−ABCD.
Use the appropriate formula from Table 1.5.1 to calculate the area of the parallelogram whose vertices are the four points P:4,13, Q:12,29, R:16,57, and S:8,41.
Use the appropriate formula from Table 1.5.1 to calculate the area of the triangle whose vertices are the three points P:1,2,3, Q:−5,3,2, and R:7,−5,4.
Prove that the point S:1,−1,4 does not lie in the plane determined by the points P, Q, and R given in Example 1.5.4.
Use the appropriate formula from Table 1.5.1 to calculate the distance of the point P:1,2,3 from the line through points Q:5,−3,7 and R:4,1,−6.
Derive the formula given in Table 1.5.1 for the distance from a point to a line.
Use the appropriate formula from Table 1.5.1 to calculate the distance of the point P:2,−3,4 to the plane through the three points Q:1,2,−3, R:5,4,7, and S:6,−5,−1.
Derive the formula given in Table 1.5.1 for the distance from a point to a plane.
Using the formulas in Table 1.5.1 for reciprocal vectors, obtain V1,V2,V3, the set of vectors reciprocal to the vectors U1=A, U2=B, U3=C, where A, B, and C are given in Example 1.5.1.
Solve nine equations in nine unknowns to find the same set of reciprocal vectors that was found in Example 1.5.10.
Solve the appropriate set of four equations in four unknowns to find V1,V2, the set of vectors reciprocal to U1=2 i−3 j, U2=3 i+4 j.
The force F=2 i−3 j+4 k is applied to the head of the position vector r=3 i+2 j−5 k.
Find τ, the torque vector, and τ, its magnitude.
What is the angle between F and r?
Find a unit vector in the direction of the axis of rotation.
If A≠0, under what conditions on B and C can A×B=A×C not imply that B=C?
If A≠0, show that A×B=A×C and A·B=A·C together imply that B=C.
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