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