Multiplication with Vectors Scalar Multiplication Dot Product Cross Product

Objectives TSW use the dot product to fin the relationship between two vectors. TSWBAT determine if two vectors are perpendicular

A bit of review A vector is a _________________ The sum of two or more vectors is called the ___________________ The length of a vector is the _____________

Find the sum Vector a = < 3, 9 > and vector b = < -1, 6 > What is the magnitude of the resultant. Hint* remember use the distance formula.

Multiplication with Vectors Scalar Multiplication Dot Product Cross Product

Scalar Multiplication: returns a vector answer Distributive Property:

Multiplication with Vectors Scalar Multiplication Dot Product Cross Product

Dot Product Given and are two vectors, The Dot Product ( inner product )of and is defined as A scalar quantity

Finding the angle between two Vectors a - b θ b

Example Find the angle between the vectors:

1:

2:

3:

Classify the angle between two vectors: Acute : ______________________________________________ Obtuse: _____________________________________________ Right: (Perpendicular , Orthogonal) _______________________

example Given three vectors determine if any pair is perpendicular THEOREM: Two vectors are perpendicular iff their Dot (inner) product is zero. Given three vectors determine if any pair is perpendicular

Ex 1:

Ex 2:

Ex 3: Find the unit vector in the same direction as v = 2i-3j-6k

Ex 4: If v = 2i - 3j + 6k and w = 5i + 3j – k Find:

Ex 5: (c) 3v (d) 2v – 3w (e)

Ex 6: Find the angle between u = 2i -3j + 6k and v = 2i + 5j - k

Ex 7: Find the direction angles of v = -3i + 2j - 6k

Any nonzero vector v in space can be written in terms of its magnitude and direction cosines as: Ex 9: Find the direction angles of the vector below. Write the answer in the form of an equation. v = 3i – 5j + 2k

We can also find the Dot Product of two vectors in 3-d space. Two vectors in space are perpendicular iff their inner product is zero.

Example Find the Dot Product of vector v and w. Classify the angle between the vectors.

Projection of Vector a onto Vector b Written :

Example: Find the projection of vector a onto vector b :

Decompose a vector into orthogonal components… Find the projection of a onto b Subtract the projection from a The projection, and a - b are orthogonal a b a-b

Multiplication with Vectors Scalar Multiplication Dot Product Cross Product

OBJECTIVE 1

OBJECTIVE 2

OBJECTIVE 3

OBJECTIVE 4

OBJECTIVE 5

Cross product Another important product for vectors in space is the cross product. The cross product of two vectors is a vector. This vector does not lie in the plane of the given vectors, but is perpendicular to each of them.

If Then the cross product of vector a and vector b is defined as follows:

The determinant of a 2 x 2 matrix

An easy way to remember the coefficients of vectors I, j, and k is to set up a determinant as shown and expand by minors using the first row. You can check your answer by using the dot product.

Example Find the cross product of vector a and vector b if: Verify that your answer is correct.

Assignment