9.7 Vectors

Finding the magnitude of a vector You begin with an initial point to a terminal point given in terms of points, usually P and Q. You graph it as you would a ray. Initial point is P(0, 0). Terminal point is Q(-6, 3). Q(-6, 3) P(0, 0)

Write the component form Here you write the following Component Form =‹x2 – x1, y2 – y1› <-6 – 0, 3 – 0> <-6, 3> is the component form. Next use the distance formula to find the magnitude. |PQ| = √(-6 – 0)2 + (3 – 0)2 = √62 + 32 = √36 + 9 = √45 ≈ 6.7 Q(-6, 3) P(0, 0)

Graph Initial/Terminal points Initial point is P(0, 2). Terminal point is Q(5, 4). Reminder that Q is the second point. P is the initial point. Graph the ray starting at P and going through Q as to the right. Then you can start looking for component form and magnitude.

Write the component form Here you write the following Component Form =‹x2 – x1, y2 – y1› <5 – 0, 4 – 2> <5, 2> is the component form. Next use the distance formula to find the magnitude. |PQ| = √(5 – 0)2 + (4 – 2)2 = √52 + 22 = √25 + 4 = √29 ≈ 5.4

Graph Initial/Terminal points Initial point is P(3, 4). Terminal point is Q(-2, -1). Reminder that Q is the second point. P is the initial point. Graph the ray starting at P and going through Q as to the right. Then you can start looking for component form and magnitude.

Write the component form Here you write the following Component Form =‹x2 – x1, y2 – y1› <-2 – 3, -1 – 4> <-5, -5> is the component form. Next use the distance formula to find the magnitude. |PQ| = √-2 – 3)2 + (-1– 4)2 = √(-5)2 + (-5)2 = √25 + 25 = √50 ≈ 7.1

Vocabulary Direction of a vector: determined by the angle, north, south, east, west Equal: same magnitude and direction Parallel: same or opposite directions

Adding Vectors Two vectors can be added to form a new vector. To add u and v geometrically, place the initial point of v on the terminal point of u, (or place the initial point of u on the terminal point of v). The sum is the vector that joins the initial point of the first vector and the terminal point of the second vector. It is called the parallelogram rule because the sum vector is the diagonal of a parallelogram. You can also add vectors algebraically.

What does this mean? Adding vectors: Sum of two vectors The sum of u = <a1,b1> and v = <a2, b2> is u + v = <a1 + a2, b1 + b2> In other words: add your x’s to get the coordinate of the first, and add your y’s to get the coordinate of the second.

Example: Let u = <3, 5> and v = <-6, -1> To find the sum vector u + v, add the x’s and add the y’s of u and v. u + v = <3 + (-6), 5 + (-1)> = <-3, 4>