Trigonometry Vector Notation

A vector is a line segment. It has a starting point and an ending point. A vector has direction and length. It may be easiest to think of a vector as a line on a road map. There are usually several ways to describe the route. Look at the diagram below.

End Point

Start Point 5 45°

Telling someone to get from the beginning to the end, you could say “Go 5 miles at an angle (in standard position) of 45 degrees.” This direct route is a vector. In this case, 5 is the magnitude, or length, of the vector, and 45° is the amplitude, or direction, of the vector.

Often, when we travel, we cannot go along the vector. Suppose you want to tell a neighbor how to go from your apartment to the library. In the diagram above, you may tell them to “Go east 3.5 miles then north 3.5 miles.” You are breaking the directions into components. We often break vectors into two components; the i component is the number of units in the x-axis direction and the j component is the number of units in the y-axis direction. In this example, we would write the vector: li=+3.5 3.5 j. The vector is the hypotenuse of the triangle; notice that the variable for the vector has an arrow symbol over it.

Vector components do not have to be positive. The vector v = −3i − 2 j , which can also be ⎛ −3⎞ written v = , is shown below. ⎝⎜ −2⎠⎟ −3i

−2 j v

The length of a vector is its magnitude. The symbol for magnitude of a vector can be written v , or like an absolute value, v , or just written “magnitude”. If you are given the components of a vector, use the distance formula to find the magnitude.

2 2 The magnitude of vector v = −3i − 2 j is ()−3 +−()2 = 13 .

Amplitude is the angle of the vector. We measure the angle as if the vector was the terminal side of an angle in standard position. When given the i and j components of the vector, use right triangle trigonometry to find the amplitude.

Example: Find the magnitude and amplitude of vector vi=12+ 5 j.

Sketch of the vector. v 5 The magnitude is ()12 2 + ()5 2 = 13 . θ 12 −1 ⎛⎞5 The amplitude is θ ==tan⎜⎟ 22.6°. ⎝⎠12

Example: Find the magnitude and amplitude of vector v = −3i − 2 j . -3 Sketch of the vector. θ

22 -2 The magnitude is ()−+−=32 () 13. v

−1 ⎛⎞−2 The amplitude is θ ==tan⎜⎟ 33.7°. ⎝⎠−3

−1 ⎛⎞−2 The angleθ ==tan⎜⎟ 33.7°, but this angle is not in standard position. From the ⎝⎠−3 diagram, we can see that the angle should be a quadrant 3 angle, 180 +=33.7 213.7

The amplitude of vector v = −3i − 2 j is 213.7°.

Try these: Sketch the following vectors in standard position. 1. f = 4i − 4 j

⎛ 1⎞ 2. g = ⎜ 7⎟ ⎝ ⎠

3. amplitude=135°, magnitude = 2 cm

Find the magnitude and amplitude of the following vectors. 4. f = 4i − 4 j

⎛ 1⎞ 5. g = ⎜ 7⎟ ⎝ ⎠

Vectors are line segments with length and direction. Compare the following pairs of vectors. Vectors v and w have exactly the same length and direction, we call them v equal vectors. They don’t need the same starting points, but they have the same w magnitude and amplitude.

v Vectors v and w have exactly the same length and start at the same point, but w they point in opposite directions, we call them opposite vectors.

v Vectors v and w have the same direction, but different lengths, we call them parallel vectors. w

Label the following pairs of vectors as equal, opposite, parallel, or none of these. (Hint- sketch the vectors first)

• a = 3i + 2 j • a = 3i + 2 j • a = 3i + 2 j ⎛ 3⎞ b =−3i − 2 j c = 6i + 4 j e = ⎜2⎟ ⎝ ⎠