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Calculus III: Section 12.1
Professor Ensley
Ship Math
September 6, 2011
Professor Ensley (Ship Math) Calculus III: Section 12.1 September 6, 2011 1 / 8
Vectors
Vectors
Professor Ensley (Ship Math) Calculus III: Section 12.1 September 6, 2011 2 / 8
Vectors
Vectors
Professor Ensley (Ship Math) Calculus III: Section 12.1 September 6, 2011 2 / 8
Vectors
Equivalent Vectors
Professor Ensley (Ship Math) Calculus III: Section 12.1 September 6, 2011 3 / 8
Vectors
Equivalent Vectors
Professor Ensley (Ship Math) Calculus III: Section 12.1 September 6, 2011 3 / 8
Vector Algebra
Magnitude of a vector
Suppose v =−→PQ with P = (a1, b1) and Q = (a2, b2), then the
magnitude or length of v is given by
‖PQ‖ =√
(a1 − a2)2 + (b1 − b2)2
So if v = 〈a, b〉, then
‖v‖ =√a2 + b2
Fact: If λ is a real number (i.e., a scalar) and v is a non-zero vector,then λv is a vector parallel to v and with magnitude
‖λv‖ = |λ|‖v‖
Professor Ensley (Ship Math) Calculus III: Section 12.1 September 6, 2011 4 / 8
Vector Algebra
Vector Algebra
Professor Ensley (Ship Math) Calculus III: Section 12.1 September 6, 2011 5 / 8
Vector Algebra
Vector Algebra
Professor Ensley (Ship Math) Calculus III: Section 12.1 September 6, 2011 5 / 8
Vector Algebra
Vector Algebra
Professor Ensley (Ship Math) Calculus III: Section 12.1 September 6, 2011 5 / 8
Vector Algebra
Vector Algebra
Professor Ensley (Ship Math) Calculus III: Section 12.1 September 6, 2011 5 / 8
Vector Algebra
Linear Combinations
A linear combination of the vectors v and w is a vector of the formu = rv + sw, where r and s are real numbers (aka scalars). We say that uis spanned by v and w in this case.
In the picture above, 〈4, 4〉 = 45〈2, 4〉+ 2
5〈6, 2〉.
Professor Ensley (Ship Math) Calculus III: Section 12.1 September 6, 2011 6 / 8
Vector Algebra
Special Vectors
A vector of length 1 is called a unit vector.
Given a vector v, the unit vector in the direction of v is denoted evand satisfies
ev =1
‖v‖v
From our study of polar coordinates, we know that
ev = 〈cos θ, sin θ〉 and so v = 〈‖v‖ cos θ, ‖v‖ sin θ〉
where θ is the angle between the vector v and the positive x-axis.
The unit vectors in the direction of the positive x and y axes aredenoted i = 〈1, 0〉 and j = 〈0, 1〉. These are called the standard basisvectors.
It is super easy to write any vector as a linear combination of thestandard basis vectors: for example, 〈3,−4〉 = 3i− 4j.
Professor Ensley (Ship Math) Calculus III: Section 12.1 September 6, 2011 7 / 8
Triangle Inequality
Triangle Inequality
Professor Ensley (Ship Math) Calculus III: Section 12.1 September 6, 2011 8 / 8