Vectors 2 3 Dot Products and Vector Projections

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Vectors 2 – 3 Dot Products and Vector Projections

Vectors 2 – 3 Dot Products and Vector Projections

Then & Now: Then: Now: Found magnitudes of and operated with algebraic vectors. 1.

Then & Now: Then: Now: Found magnitudes of and operated with algebraic vectors. 1. Find the dot product of two vectors. 2. Find the projection of one vector onto another.

Dot product of Vectors in a Plane

Dot product of Vectors in a Plane

Example 1: Find the dot product of u and v. Then determine if u

Example 1: Find the dot product of u and v. Then determine if u and v are orthogonal. u = < 3 , 6 > and v = < - 4, 2 >

Example 1: Find the dot product of u and v. Then determine if u

Example 1: Find the dot product of u and v. Then determine if u and v are orthogonal. u = < 3 , 6 > and v = < - 4, 2 >

Example 1: Find the dot product of u and v. Then determine if u

Example 1: Find the dot product of u and v. Then determine if u and v are orthogonal. u = < 3 , 6 > and v = < - 4, 2 >

Example 1: Find the dot product of u and v. Then determine if u

Example 1: Find the dot product of u and v. Then determine if u and v are orthogonal. u = < 3 , 6 > and v = < - 4, 2 >

Example 1: Find the dot product of u and v. Then determine if u

Example 1: Find the dot product of u and v. Then determine if u and v are orthogonal. u = < 3 , 6 > and v = < - 4, 2 >

Example 2: Find the dot product of u and v. Then determine if u

Example 2: Find the dot product of u and v. Then determine if u and v are orthogonal. u = < 2 , 5 > and v = < 8, 4 >

Example 2: Find the dot product of u and v. Then determine if u

Example 2: Find the dot product of u and v. Then determine if u and v are orthogonal. u = < 2 , 5 > and v = < 8, 4 >

Example 3: Find the dot product of u and v. Then determine if u

Example 3: Find the dot product of u and v. Then determine if u and v are orthogonal. u = < 3 , - 2> and v = < - 5, 1 >

Example 3: Find the dot product of u and v. Then determine if u

Example 3: Find the dot product of u and v. Then determine if u and v are orthogonal. u = < 3 , - 2> and v = < - 5, 1 >

Example 4: Find the dot product of u and v. Then determine if u

Example 4: Find the dot product of u and v. Then determine if u and v are orthogonal. u = < - 2 , - 3 > and v = < 9, - 6 >

Example 4: Find the dot product of u and v. Then determine if u

Example 4: Find the dot product of u and v. Then determine if u and v are orthogonal. u = < - 2 , - 3 > and v = < 9, - 6 >

Properties of the Dot Product Commutative Property Distributive Property Scalar Multiplication Property Zero Vector

Properties of the Dot Product Commutative Property Distributive Property Scalar Multiplication Property Zero Vector Dot Product Property Dot Product and Vector Magnitude Relationship

Example 5: Use the dot product to find the magnitude of a = <

Example 5: Use the dot product to find the magnitude of a = < - 5, 12 >.

Example 5: Use the dot product to find the magnitude of a = <

Example 5: Use the dot product to find the magnitude of a = < - 5, 12 >.

Example 5: Use the dot product to find the magnitude of a = <

Example 5: Use the dot product to find the magnitude of a = < - 5, 12 >.

Example 5: Use the dot product to find the magnitude of a = <

Example 5: Use the dot product to find the magnitude of a = < - 5, 12 >.

Example 5: Use the dot product to find the magnitude of a = <

Example 5: Use the dot product to find the magnitude of a = < - 5, 12 >.

Example 5: Use the dot product to find the magnitude of a = <

Example 5: Use the dot product to find the magnitude of a = < - 5, 12 >.

Example 6: Use the dot product to find the magnitude of a = <

Example 6: Use the dot product to find the magnitude of a = < 12, 16 >.

Example 6: Use the dot product to find the magnitude of a = <

Example 6: Use the dot product to find the magnitude of a = < 12, 16 >.

Example 7: Use the dot product to find the magnitude of a = <

Example 7: Use the dot product to find the magnitude of a = < - 1, - 7 >.

Example 7: Use the dot product to find the magnitude of a = <

Example 7: Use the dot product to find the magnitude of a = < - 1, - 7 >.

Vector Projection

Vector Projection

Example 8: Find the vector projection of u = < 3, 2 > onto

Example 8: Find the vector projection of u = < 3, 2 > onto v = < 5, - 5 >.

Example 8: Find the vector projection of u = < 3, 2 > onto

Example 8: Find the vector projection of u = < 3, 2 > onto v = < 5, - 5 >.

Example 8: Find the vector projection of u = < 3, 2 > onto

Example 8: Find the vector projection of u = < 3, 2 > onto v = < 5, - 5 >.

Example 8: Find the vector projection of u = < 3, 2 > onto

Example 8: Find the vector projection of u = < 3, 2 > onto v = < 5, - 5 >.

Example 8: Find the vector projection of u = < 3, 2 > onto

Example 8: Find the vector projection of u = < 3, 2 > onto v = < 5, - 5 >.

Example 8: Find the vector projection of u = < 3, 2 > onto

Example 8: Find the vector projection of u = < 3, 2 > onto v = < 5, - 5 >.

Example 8: Find the vector projection of u = < 3, 2 > onto

Example 8: Find the vector projection of u = < 3, 2 > onto v = < 5, - 5 >.

Example 8: Find the vector projection of u = < 3, 2 > onto

Example 8: Find the vector projection of u = < 3, 2 > onto v = < 5, - 5 >.

Example 9: Find the vector projection of u = < 1, 2 > onto

Example 9: Find the vector projection of u = < 1, 2 > onto v = < 8, 5 >.

Example 9: Find the vector projection of u = < 1, 2 > onto

Example 9: Find the vector projection of u = < 1, 2 > onto v = < 8, 5 >.

Example 10: Find the vector projection of u = < 4, -3 > onto

Example 10: Find the vector projection of u = < 4, -3 > onto v = < 2, 6 >.

Example 10: Find the vector projection of u = < 4, -3 > onto

Example 10: Find the vector projection of u = < 4, -3 > onto v = < 2, 6 >.

Example 11: Find the vector projection of u = < -3, 4 > onto

Example 11: Find the vector projection of u = < -3, 4 > onto v = < 6, 1 >.

Example 11: Find the vector projection of u = < -3, 4 > onto

Example 11: Find the vector projection of u = < -3, 4 > onto v = < 6, 1 >.

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