Chapter 1: Vectors, Lines and Planes
Section 1.5: Applications of Vector Products
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Essentials
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Table 1.5.1 lists formulae that combine both dot and cross products.
Triple Scalar (Box) Product
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Area of parallelogram with edges A and B
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Area of triangle with edges A and B
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Volume of parallelepiped with edges A, B, and C
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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
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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
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Vectors and are reciprocal sets of vectors if
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, ,
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Torque exerted about O by a force F acting at the head of , a vector of length with tail at O
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Table 1.5.1 Formulas involving dot and cross products
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Examples
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Example 1.5.1
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If , , and ,
a)
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Compute , the Triple Scalar (or Box) Product .
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b)
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Verify the identity for the Triple Scalar Product.
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Example 1.5.2
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For the vectors A, B, and C of Example 1.5.1, and ,
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Example 1.5.3
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Use the appropriate formula from Table 1.5.1 to calculate the area of the parallelogram whose vertices are the four points P:, Q:, R:, and S:.
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Example 1.5.4
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Use the appropriate formula from Table 1.5.1 to calculate the area of the triangle whose vertices are the three points P:, Q:, and R:
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Example 1.5.5
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Prove that the point S: does not lie in the plane determined by the points P, Q, and R given in Example 1.5.4.
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Example 1.5.6
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Use the appropriate formula from Table 1.5.1 to calculate the distance of the point P: from the line through points Q: and R:.
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Example 1.5.7
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Derive the formula given in Table 1.5.1 for the distance from a point to a line.
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Example 1.5.8
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Use the appropriate formula from Table 1.5.1 to calculate the distance of the point P: to the plane through the three points Q:, R:, and S:
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Example 1.5.9
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Derive the formula given in Table 1.5.1 for the distance from a point to a plane.
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Example 1.5.10
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Using the formulas in Table 1.5.1 for reciprocal vectors, obtain , the set of vectors reciprocal to the vectors , , , where A, B, and C are given in Example 1.5.1.
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Example 1.5.11
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Solve nine equations in nine unknowns to find the same set of reciprocal vectors that was found in Example 1.5.10.
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Example 1.5.12
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Solve the appropriate set of four equations in four unknowns to find , the set of vectors reciprocal to , .
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Example 1.5.13
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The force is applied to the head of the position vector .
a)
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Find , the torque vector, and , its magnitude.
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b)
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What is the angle between F and r?
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c)
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Find a unit vector in the direction of the axis of rotation.
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Example 1.5.14
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If , under what conditions on B and C can not imply that ?
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Example 1.5.15
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If , show that and together imply that .
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