Review 11.2 Vectors in space

Basic Information: Component form: v = <v1, v2, v3> Unit vector form: v = v1i + v2j + v3k Standard unit vector notation uses these unit vectors: i = <1, 0, 0>, j = <0, 1, 0> and k = <0, 0, 1>. Zero vector: 0 = <0, 0, 0>

Vectors in Space: 1. Two vectors are equal if and only if their corresponding components are equal. 2. The length of vector u = <u1, u2, u3> is: ||u|| = √(u1 3. A unit vector u in the direction of v is given by: u = v / ||v||, v ≠ 0 4. The sum of u = <u1, u2, u3> and v = <v1, v2, v3> is: u + v = <u1 + v1, u2 + v2, u3 + v3> 5. The scalar multiple of the real number c and vector u is: cu = <cu1, cu2, cu3> 6. The dot product of vector u and vector v is: u • v = u1v1 + u2v2 + u3v3

Angle Between Two Nonzero Vectors If θ is the angle between two nonzero vectors u and v, then: cosθ = (u • v) / (||u|| ||v||) If the dot product of two nonzero vectors is 0, the vectors are orthogonal (perpendicular). In general, two nonzero vectors u and v are parallel if there is some scalar c such that u = cv.

Using Vectors to Determine Collinear Points The points P, Q, and R are collinear if and only if the vectors PQ and PR are parallel.

Example #1:

Find the dot product of the vectors: u = <5, 8, 2> and v = <7, 1, 6>.

u • v = (5)(7) + (8)(1) + (2)(6) = 35 + 8 + 12 = 55