SECTION 9.5 The Dot Product 655 and v ,2 which is orthogonal to w . See Figure 73(a) and (b).The vector v1 is called the vector projection of v onto w . The vector v1 is obtained by dropping a perpendicular from the terminal point of v to the line containing w . The vector v1 is the vector from P to the intersection of the line containing w and the perpendicular. The vector v2 is given by v v v . 2 1 = − Note that v v v , 1 2 = + the vector v1 is parallel to w , and the vector v2 is orthogonal to w . This is the decomposition of v that was sought. Now we seek a formula for v1 that is based on a knowledge of the vectors v and w . Since v v v , 1 2 = + we have v w v v w v w v w 1 2 1 2 ( ) ⋅ = + ⋅ = ⋅ + ⋅ (9) Since v2 is orthogonal to w , we have v w 0. 2 ⋅ = Since v1 is parallel to w , we have v w 1 α = for some scalar .α Equation (9) can be written as v w w w w v w w 2 2 α α α ⋅ = ⋅ = = ⋅ α= ⋅ = v w; v w 0 1 2 Then v w v w w w 1 2 α = = ⋅ Figure 73 P v w v1 v2 (a) v w v1 v2 P (b) Figure 74 x y w 5 i 1 j v 5 i 1 3j v2 5 2i 1 j v1 5 2(i 1 j) THEOREM If v and w are two nonzero, nonorthogonal vectors, the vector projection of v onto w is v v w w w 1 2 = ⋅ (10) The decomposition of v into v1 and v ,2 where v1 is parallel to w , and v2 is orthogonal to w , is v v w w w v v v 1 2 2 1 = ⋅ = − (11) Decomposing a Vector into Two Orthogonal Vectors Find the vector projection of v i j3 = + onto w i j. = + Decompose v into two vectors, v1 and v ,2 where v1 is parallel to w , and v2 is orthogonal to w . Solution EXAMPLE 5 Use formulas (10) and (11). v v w w w w w i j v v v i j i j i j 1 3 2 2 2 3 2 1 2 2 2 1 ( ) ( ) ( ) ( ) = ⋅ = + = = + = − = + − + = − + See Figure 74. Now Work PROBLEM 21 Finding the Force Required to Hold a Wagon on a Hill A wagon with two small children as occupants weighs 100 pounds and is on a hill with a grade of 20 .° What is the magnitude of the force that is required to keep the wagon from rolling down the hill? EXAMPLE 6 (continued)
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