Emily Forward, RN, Margaret Gerulski, RN, Mary Jacobs, RN, Michele Niles, RN, & Cheryl Rose, RN.
Section 5.1: LENGTH AND DOT PRODUCT IN Rn
Transcript of Section 5.1: LENGTH AND DOT PRODUCT IN Rn
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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 .