Education.com
Try
Brainzy
Try
Plus

# Vectors and Vector Arithmetic for AP Calculus

(not rated)
By McGraw-Hill Professional
Updated on Oct 24, 2011

Practice problems for these concepts can be found at:  Graphs of Functions and Derivatives Practice Problems for AP Calculus

### Vectors

A vector represents a displacement of both magnitude and direction. The length, r, of the vector is its magnitude, and the angle, θ, it makes with the x -axis gives its direction. The vector can be resolved into a horizontal and a vertical component.

A unit vector is a vector of magnitude 1. If i = <1, 0> is the unit vector parallel to the positive x -axis, that is, a unit vector with direction angle θ =0, and j = <0, 1> is the unit vector parallel to the y -axis, with an angle , then any vector in the plane can be represented as xi +y j or simply as the ordered pair

#### Example 1

Find the magnitude and direction of the vector represented by .

Step 1: Calculate the magnitude

Step 2: The terminal point of the vector is in the fourth quadrant. Calculate –.464 radians. This angle falls in quadrant IV.

#### Example 2

Find the magnitude and direction of the vector represented by .

Step 1: Calculate the magnitude

Step 2: The terminal point of the vector is in the third quadrant. Calculate

#### Example 3

Find the magnitude and direction of the vector represented by

Step 1: Calculate the magnitude

Step 2: The terminal point of the vector is in the second quadrant. Calculate

#### Example 4

Find the ordered pair representation of a vector of magnitude 12 and direction x =12 cos

### Vector Arithmetic

If C is a constant, r1 = x1, y1 and r2 = x2, y2, then:

Addition: r1 +r2 = x1 +x2, y1 + y2
Subtraction: r1r2 = x1x2, y1y2
Scalar Multiplication: Cr1 = Cx1, Cy1 Note: ||Cr1|| = ||C|| · ||r1||
Dot Product: The dot product of two vectors is r1 · r2 =||r1|| · ||r2|| · cos θ or r1 · r2 =x1x2 + y1 y2.

#### Parallel and Perpendicular Vectors

If r2 =Cr1, then r1 and r2 are parallel.

If r1 · r2 =0, then r1 and r2 are perpendicular or orthogonal.

The angle between two vectors can be found by cos

#### Example 1

Given r1 = 4, –7, = r2 = –3, –2 and r2 = –1, 5, find 3r1 – 5r2 +2r3. 3r1 – 5r2 – 2r3 =34, –7 – 5–3, –2+2–1, 5= 12, –21–15, –10+ –2, 10= 27, –11–2, 10 = 29, –21.

#### Example 2

Determine whether the vectors r1 = 4, –7 and r2 = –3, –2 are orthogonal. If the vectors are not orthogonal, approximate the angle between them.

Step 1. Find the dot product r1 · r2 = 4(–3)+(–7)(–2)=2. Since the dot product is not equal to zero, the vectors are not orthogonal.

Step 2. If θ is the angle between the vectors, then cos θ = The dot product is 2, ||r1 = ≈ 0.0688 and θ ≈ 1.5019 radians.

Practice problems for these concepts can be found at:  Graphs of Functions and Derivatives Practice Problems for AP Calculus

150 Characters allowed

### Related Questions

#### Q:

See More Questions

### Today on Education.com

Top Worksheet Slideshows