1. Vectors in Space 11.2 JMerrill, 2010

2. Rules The same rules apply in 3-D space: The component form is found by subtracting the coordinates of the initial point from the corresponding coordinates of the terminal point. Two vectors are = iff their corresponding components are =. The magnitude (length) of v Vector addition still means you add the value in the x place, the y place, and the z place

3. Scalar multiplication still means you distribute the scalar over the component form The dot product of If the dot product = 0, then the vectors are orthogonal. The angle between two nonzero vectors u and v is

4. Finding the Component Form of a Vector Find the component form and magnitude of vector v having an initial point of (3,4,2) and a terminal point of (3,6,4) The component form is The magnitude is

5. Finding the Dot Product of Two Vectors Find the dot product of and Remember the dot product is a scalar, not a vector.

6. Find the Angle Between Two Vectors Find the angle between u = and v =

7. Parallel Vectors Vector w has an initial point (1,-2,0) and a terminal point (3,2,1). Which of the following vectors is parallel to w? A. u = <4,8,2> B. v = <4,8,4> Put w into component form <2,4,1> Vector u is the answer because it is just double vector w. U can be written as 2<2,4,1>

8. Using Vectors to Determine Collinear Points Determine whether the points P(2,-1,4), Q(5,4,6), and R(-4,-11,0) are collinear. The points P,Q, and R are collinear iff the vectors PQ and PR are parallel. Because PR = -2PQ, the vectors are parallel. Therefore, the points are collinear.

9. Finding the Terminal Point of a Vector The initial point of v = is P(3,-1,6). What is the terminal point? We use the initial point of P(3,-1,6) and the terminal point of Q(q 1, q 2, q 3 ) PQ = = The 1 st term = the 1 st term, the 2 nd term = the 2 nd term… So, q 1 –3 = 4, q 2 -(-1) = 2, q 3 –6 = -1 Point Q = (7, 1, 5)