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MAT 1236 12.3 Handout The Dot Product a b a1 , a2 , a3 b1 , b2 , b3 a1b1 a2b2 a3b3 Example 1 (a) 1,0, 1 0,1, 1 (b) 2i 3 j k i k Dot product and Length a a a1 , a2 , a3 a1 , a2 , a3 Properties Geometric Meaning For 0 , 1 Formula cos a b a b Example 2 a 1, 1,1 , b 1, 1, 2 cos a b a b Orthogonal Vectors Two vectors are orthogonal if the angle between them is a right angle. Example 3 Find the value of x such that the given vectors are orthogonal. a x, x, 1 , b 1, x,6 2 Projection Vector Projection of b onto a proja b a b a 2 a Scalar Projection of b onto a compab a b a Q: How do I know proja b is in the same direction of a ? Q: Does the length of the vector proja b agree with what we know? 3 Proof A unit vector in the direction of x and a is So, according to the formula from the last slide, we have x x We know x compab a b . Thus, a x Example 4 Find the vector projection of b onto a. a 2,3 , b 4,1 proja b a b a 2 a 4 Classwork 1. Find a unit vector that is orthogonal to both i j and 2i j k . Solution Suppose the required unit vectors are of the form i j k for some numbers , , and . Step 1 Write down the condition for which i j k is orthogonal to i j . i j k i j 0 0 Step 2 Write down the condition for which i j k is orthogonal to 2i j k . Step 3 Write down the condition for which i j k is a unit vector. 5 Step 4 Solve the 3 equations in the above steps for , , and . (Be sure to write down necessary connection statements.) Answers 1 3 3 3 3 3 3 i j k i j k OR 3 3 3 3 3 3 6