Section 12.3 The Dot Product

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Presentation transcript:

Section 12.3 The Dot Product MAT 1236 Calculus III Section 12.3 The Dot Product http://myhome.spu.edu/lauw

HW… WebAssign 12.3 (19 problems, 108 min.)

Preview Define a new operation on vectors. Angle(s) between vectors Projection of vectors Formula Why the formula makes sense? (More important in a long run)

We are Interested in … The angle between two forces The component of one force along the direction of another force

Vectors The angle between two forces

Dot Product If and , the dot product of a and b is the number

Example 1 (a) (b)

Dot product and Length

Properties

Properties

Geometric Meaning

Formula

PPFNE Derive the formula of the angle between two vectors

Example 2 Find the angle between the given vectors

Special Case: Orthogonal Vectors Two vectors are orthogonal if the angle between them is a right angle.

Example 3 Find the value of x such that the given vectors are orthogonal.

Projection Vector Projection Scalar Projection

Remark Scalar Projection

Make Sense? Vector Projection How do I know the vector is in the same direction of a?

Make Sense? Vector Projection How do I know <2,2> is in the same direction of <1,1>?

Make Sense? Vector Projection Does the “length” of the vector agree with what we know?

Proof: We need to recall… Vector Projection

Unit Vectors A unit vector is a vector with length is 1 e.g. 1, 0 , 0, 1 , 2 2 , 2 2

Unit Vectors The unit vector 𝑢 in the same direction as vector 𝑏 𝑢= 1 𝑏 𝑏

Unit Vectors Unit vector 𝑢 in the same direction as vector 𝑏 𝑢= 1 𝑏 𝑏 So, 𝑏= 𝑏 𝑢 A vector equals to its length times the unit vector along the same direction

Proof:

Example 4 Find the vector projection of 𝑏 onto 𝑎.