Vectors in the Plane

1. Vectors in the Plane Section 10.2a

2. Vectors in the Plane Some quantities only have magnitude, and are called scalars … Examples? Some quantities have both magnitude and direction, and can be represented by directed line segments… Examples? The directed line segment has initial point A and terminal point B; its length is denoted by . Directed line segments that have the same length and direction are equivalent. Terminal point Initial point B A

3. Vectors in the Plane Definitions: Vector, Equal Vectors A vector in the plane is represented by a directed line segment. Two vectors are equal (or the same) if they have the same length and direction. Definition: Component Form of a Vector If v is a vector in the plane equal to the vector with initial point and terminal point , then the component form of v is v The magnitude (length) of v is

4. Finding Component Form Find the (a) component form and (b) length of the vector with initial point P = (–3, 4) and terminal point Q = (–5, 2). Start with a graph of the vector. (a) (b) If a vector has a magnitude of 1, then it is a unit vector. The slope of a nonvertical vector is the slope shared by the lines parallel to the vector. The zero vector: 0 is the only vector with no direction.

5. Vectors in the Plane Definitions: Vector Operations Let , be vectors with k a scalar (real number). Addition: Subtraction: Scalar Multiplication: Negative (opposite):

6. Vectors in the Plane Vector Addition When adding vectors geometrically, align them “head to tail,” and the sum is called the resultant vector. This geometric description of vector addition is sometimes called the parallelogram law: u v u+ v v u

7. Vectors in the Plane Scalar Multiplication When multiplying scalars and vectors geometrically, the scalar simply stretches (k > 1) or shrinks (k < 1) the vector. If k is negative, the vector also changes to the opposite direction: 2u u –2u 0.7u

8. Vectors in the Plane Properties of Vector Operations Let u, v, and w be vectors and a, b be scalars. 1. 2. 3. 4. 5. 6. 7. 8. 9.

9. Vectors in the Plane Definition: Dot Product (Inner Product) The dot product (or inner product) u v (“u dot v”) of vectors and is the number Definition: Angle Between Two Vectors The angle between nonzero vectors u and v is

10. Practice Problems Find the component form of the vector v of length 3 that makes an angle of with the positive x-axis. Let and . Find:

11. Practice Problems Find the measure of angle C in the triangle ABC defined by the following points: Sketch a graph of the triangle. The angle is formed by vectors and . Component forms of these vectors: Angle between these vectors:

12. Practice Problems Find the measure of angle C in the triangle ABC defined by the following points:

13. Practice Problems Find the measure of angle C in the triangle ABC defined by the following points:

14. Practice Problems Find a unit vector in the direction of the given vector. Since a unit vector has a magnitude of 1, simply divide the given vector by its own magnitude (this will keep the direction the same but stretch or shrink the vector to the correct length). Unit Vector: How can we verify this answer?