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Section 5.1: LENGTH AND DOT PRODUCT IN nR

When you are done with your homework you should be able to…

Find the length of a vector and find a unit vector Find the distance between two vectors Find a dot product and the angle between two vectors, determine

orthogonality, and verify the Cauchy-Schwartz Inequality, the triangle inequality, and the Pythagorean Theorem

Use a matrix product to represent a dot product

DEFINITION OF LENGTH OF A VECTOR IN nR

The __________________, or _____________________ of a vector 1 2, , ..., nv v vv in ______ is given by

When would the length of a vector equal to 0?

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Example 1: Consider the following vectors:

11,2

u 12,2

v

a. Find u

b. Find v

c. Find u v

d. Find 3u

e. Any observations?

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THEOREM 5.1: LENGTH OF A SCALAR MULTIPLE

Let v be a vector in nR and let c be a scalar. Then

where _____ is the _____________ ______________ of c .

Proof:

THEOREM 5.2: UNIT VECTOR IN THE DIRECTION OF v

If v is a nonzero vector in nR , then the vector

has length _____ and has the same ________________ as v .

Proof:

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Example 2: Find the vector v with 3v and the same direction as 0, 2,1, 1 u .

DEFINITION OF DISTANCE BETWEEN TWO VECTORS

The distance between two vectors u and v in nR is

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Example 3: Find the distance between 1,1,2u and 1,3,0 v .

DEFINITION OF DOT PRODUCT IN nR

The dot product of 1 2, ,..., nu u uu and 1 2, ,..., nv v vv is the _____________ quantity

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Example 4: Consider the following vectors:

1,2 u 2, 2 v

a. Find u v

b. Find v v

c. Find 2u

d. Find u v v

e. Find 5u v

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THEOREM 5.3: PROPERTIES OF THE DOT PRODUCT

If u , v and w are vectors in nR , and c is a scalar, then the following properties are true.

1. _________________ u v

2. _________________ u v w

3. __________ __________c u v

4. _________________ v v

5. 0, and 0 iff ____________ . v v v v

Example 5: Find 3 3 u v u v given that 8 u u , 7 u v , and 6 v v .

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THEOREM 5.4: THE CAUCHY-SCWARZ INEQUALITY

If u and v are vectors in nR , then

where ____________ denotes the _______________ value of u v .

Proof:

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Example 6: Verify the Cauch-Schwarz Inequality for 1,0 u and 1,1v .

DEFINITION OF THE ANGLE BETWEEN TWO VECTORS IN nR

The ____________ _____ between two nonzero vectors in nR is given by

Example 6: Find the angle between 2, 1 u and 2,0v .

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DEFINITION OF ORTHOGONAL VECTORS

Two vectors u and v in nR are orthogonal if

Example 7: Determine all vectors in 2R that are orthogonal to 3,1u .

THEOREM 5.5: THE TRIANGLE INEQUALITY

If u and v are vectors in nR , then

Proof:

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THEOREM 5.6: THE PYTHAGOREAN THEOREM

If u and v are vectors in nR , then u and v are orthogonal if and only if

Example 8: Verify the Pythagoren Theorem for the vectors 1,1 u and

2,0v .