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Page 1 Math 251-copyright Joe Kahlig, 15A Section 11.2: Vectors and the Dot Product in Three Dimensions Definition: A three dimensional vector is an ordered triple a = ha1 , a2 , a3 i of real numbers. − − → Definition: Given the points A(x1 , y1 , z1 ) and B(x2 , y2 , z2 ), the vector a with representation AB is a = hx2 − x1 , y2 − y1 , z2 − z1 i − −→ − −→ Example: For the points, A(1, 2, 8) and B(4, 7, 2), find AB and BA. Definition: Let a = ha1 , a2 , a3 i, b = hb1 , b2 , b3 i and c be a scalar, i.e. c ∈ ℜ. Scalar Multiplication: ca = c ha1 , a2 , a3 i = hca1 , ca2 , ca3 i Length or magnitude of a is |a| = q a21 + a22 + a23 Vector Addition: a + b = ha1 + b1 , a2 + b2 , a3 + b3 i Zero vector: 0 = h0, 0, 0i and |0| = 0 Definition: Two vectors are parallel if one vector is a scalar multiple of the other. i.e. there exists a c ∈ ℜ such that ca = b. Definition: A vector of length 1 is called a unit vector. To find a unit vector in the same direction as a, divide vector a by its magnitude. This process is called normalizing a. Example: Find the following using these vectors: a = h1, 2, 4i, b = h3, 1, −2i, and c = h2, −4, 1i. A) 2a + 3b = B) a − 2c = C) Find a vector of length 3 in the opposite direction of a. Math 251-copyright Joe Kahlig, 15A Page 2 Definition: The dot product of two nonzero vectors a and b is the number a · b = |a||b| cos θ where θ is the angle between the vectors a and b, 0 ≤ θ ≤ π. If either a or b is 0, then a · b = 0. The dot product of a = ha1 , a2 , a3 i and b = hb1 , b2 , b3 i is a · b = a1 b1 + a2 b2 + a3 b3 Definition: Two non-zero vectors a and b are orthogonal if and only if a · b = 0, i.e. the angle between them is π/2. Example: Find the following using these vectors: a = h−1, −2, −3i, b = h−10, 2, 1i, and c = h2, 8, −6i. A) a · b = B) a · c = C) Find the angle between a and b. Example: If |a| = 1 and |b| = 2, what is the maximum for a · b? What does this say about the vectors? Page 3 Math 251-copyright Joe Kahlig, 15A Directional angles/and Direction Cosines Example: Find the direction angles for a = h1, 0, 5i Projections Scalar projection of b onto a: compa b = a·b |a| Vector projection of b onto a: proja b = a·b a |a|2 Example: Find the vector and scalar projections of m = h2, 1, 5i onto n = h1, 2, 3i