week3(1) - Lecture 9 Cross product triple products and...

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Lecture 9 Cross product, triple products and applications. Two non-zero vectors u and v are parallel if u = λ v for a scalar λ 2 R . They are perpendicular , or orthogonal , if u · w = 0 , hence the angle between them is / 2 . The vector projection of u onto v is proj v u = u · v v · v v = ( u · ˆv ) ˆv where ˆv is a unit vector parallel to v . The vector component of u orthogonal to v is u - proj v u . — I.9.1—
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Example 9.1 (a) Find the vector projection of (1 , 2 , 3) in the direction of (1 , 0 , 0) . (b) Find the vector projection of (1 , 2 , 3) in the direction of (1 , 1 , 1) . (b) Find the vector component of (1 , 2 , 3) orthogonal to (1 , 1 , 1) . — I.9.2—
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Cross Product also known as Vector Product . Let u = ( u 1 , u 2 , u 3 ) and v = ( v 1 , v 2 , v 3 ) . The cross product of u and v is algebraically defined u v = ( u 2 v 3 - u 3 v 2 ) i + ( u 3 v 1 - u 1 v 3 ) j + ( u 1 v 2 - u 2 v 1 ) k Geometrically , the cross product of u and v is u v = || u || || v || sin ˆn where 0 and ˆn is a unit vector perpendicular to both u and v using the right-hand rule. Example 9.2 (1 , 0 , 0) (0 , 1 , 0) = — I.9.3—
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Task: Find a vector that is perpendicular to both u and v . Picture this: What are u · ( u v ) and v · ( u v ) ? Later we will use this fact to define a plane. — I.9.4—
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Example 9.3 Find a vector that is perpendicular to (1 , 0 , - 1) and (1 , 2 , 1) . — I.9.5—
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It is more convenient to remember the cross product as a deter- minant. u v = i j k u 1 u 2 u 3 v 1 v 2 v 3 Use cofactor expansion along first row! Example 9.4 (2 , 3 , 1) (1 , 1 , 1) = — I.9.6—
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When is u v = 0 ? Let u and v be vectors in R 3 . What is the area of the triangle with sides given by u and v ? Example 9.5 Find the area of the triangle with vertices (2 , - 5 , 4) , (3 , - 4 , 5) and (3 , - 6 , 2) . — I.9.7—
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Lagrange’s identity: || u v || 2 = The scalar triple product is given by ( u v ) · w . Note that ( u v ) · w = ( w u ) · v = - ( u w ) · v = . . . Since u v = and w = w 1 i + w 2 j + w 3 k ( u v ) · w = | ( u v ) · w | gives the volume of a solid shape. — I.9.8—
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Example 9.6 Find the volume of the solid shape (parallelepiped) with adjacent edges --! PQ , -! PR and -! PS where the points are P (2 , - 1 , 1) , Q (4 , 6 , 7) , R (5 , 9 , 7) and S (8 , 8 , 8) .
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