Vectors Are Quantities That Are Described by Both Magnitude and Direction

Ch7.4 Vectors.

Vectors are quantities that are described by both magnitude and direction.

The magnitude of , denoted , is the length of the line segment.

Ex 1. Vector v extends from (0, 0) to (−4, 2). Vector w extends from (1, 3) to (−3, 5).

a)  Find the magnitude of v and the magnitude of w.

b)  Are v and w equal?

Ex 2. Vector v extends from (0, 0) to (3, 4). Vector w extends from (−2, 3) to (−5, −1).

a)  Find the magnitude of v and magnitude of w.

b)  Are v and w equal?

Component Form of a Vector

Let P(x1, y1) and Q(x2, y2) be points in the plane, and let . The component form of v is given by

Ex 3. Suppose that vector v has initial point (−2, 1) and terminal point (2, −5).

a)  Find the component form of v.

b)  Sketch v in standard position.

c)  Sketch v with its initial point at (2, 3).

Ex 4. Suppose that vector v has initial point (−4, 5) and terminal point (−1, 12).

a)  Find the component form of v.

b)  If v is placed with initial point at (−2, −4), what is the terminal point of v?

Equality of Vectors

If v = 〈a1, b1〉 and w = 〈a2, b2〉, then v = w if and only if a1 = a2 and b1 = b2.

Operations on Vectors

Let v = 〈a1, b1〉, w = 〈a2, b2〉, and c be a real number.

Vector addition: v + w = 〈a1 + a2, b1 + b2〉

Vector subtraction: v − w = 〈a1 − a2, b1 − b2〉

Multiplication of a vector by a scalar: cv = 〈ca1, cb1〉

Ex 5. Given v = 〈1, 4〉 and w = 〈3, −2〉 find

a)  v + w

b)  v – w

Ex 6. Given r = 〈−3, 8〉 and s = 〈2, −5〉, find

a)  5r

b)  2r − 3s

Find a Unit Vector in the Direction of a Given Vector

If v = 〈a, b〉, then a unit vector uv in the direction of v is given by

Ex 7. Find a unit vector in the direction of

a)  w = 〈−3, 2〉. b) v = 〈5, −1〉.

Represent Vectors in Terms of i and j

The representation of v = 〈a, b〉 in terms of i and j is v = ai + bj.

The values a and b are called the scalar horizontal and vertical components of v, respectively.

Ex 8. Writing Vectors in Terms of i and j

a)  Write 〈−3, 5〉 in terms of i and j.

b)  Given w = 6.2i − 3.4j and v = −1.7i + 2.2j, write 3w − 5v in terms of i and j.

Magnitude, Direction, and Components of a Vector

Let v = 〈a, b〉 be a vector in standard position, and let 0° ≤ θ < 360° be the direction of v measured counterclockwise from the positive x-axis.

·  and (magnitude and direction of v)

·  a = ‖v‖ cos θ and b = ‖v‖ sin θ

·  v = 〈a, b〉 = 〈‖v‖ cos θ, ‖v‖ sin θ〉 or v = ai + bj = ‖v‖ cos θi + ‖v‖ sin θj

Ex 9. A force of 100 lb is applied to a hook on the ceiling at an angle of 45° with the horizontal. Write the force vector F in terms of i and j.