Consider the vectors u = ( − 2 , 2 , − 3 ) u=(-2,2, - 3) u = ( − 2 , 2 , − 3 ) , v = ( − 1 , 3 , − 4 ) , w = ( 2 , − 6 , 2 ) v=(-1,3, - 4), w=(2, - 6,2) v = ( − 1 , 3 , − 4 ) , w = ( 2 , − 6 , 2 ) and the points A ( 2 , 6 , − 1 ) A (2,6,-1) A ( 2 , 6 , − 1 ) and B ( − 3 , − 5 , 7 ) B(-3, - 5,7) B ( − 3 , − 5 , 7 ) .
a) The distance between the two points is equal to ( − 3 − 2 ) 2 + ( − 5 − 6 ) 2 + ( 7 − ( − 1 ) ) 2 = 25 + 121 + 64 = 210 ≈ 14.49 \sqrt{(-3-2)^2+(-5-6)^2+(7-(-1))^2}=\sqrt{25+121+64}=\sqrt{210}\approx14.49 ( − 3 − 2 ) 2 + ( − 5 − 6 ) 2 + ( 7 − ( − 1 ) ) 2 = 25 + 121 + 64 = 210 ≈ 14.49
b.) ∣ ∣ 2 u − 3 v + 1 2 w ∣ ∣ = ∣ ∣ 2 ( − 2 , 2 , − 3 ) − 3 ( − 1 , 3 , − 4 ) + 1 2 ( 2 , − 6 , 2 ) ∣ ∣ = ∣ ∣ ( 0 , − 8 , 7 ) ∣ ∣ = ( − 8 ) 2 + 7 2 = 113 ≈ 10.63 ||2u - 3v + \frac{1}{2}w|| =||2(-2,2,-3)-3(-1,3,-4)+\frac{1}{2}(2,-6,2)||=||(0,-8,7)||=\sqrt{(-8)^2+7^2}=\sqrt{113}\approx 10.63 ∣∣2 u − 3 v + 2 1 w ∣∣ = ∣∣2 ( − 2 , 2 , − 3 ) − 3 ( − 1 , 3 , − 4 ) + 2 1 ( 2 , − 6 , 2 ) ∣∣ = ∣∣ ( 0 , − 8 , 7 ) ∣∣ = ( − 8 ) 2 + 7 2 = 113 ≈ 10.63
с) The unit vector in the direction of w w w is w ∣ ∣ w ∣ ∣ = ( 2 , − 6 , 2 ) 4 + 36 + 4 = ( 2 , − 6 , 2 ) 44 = 2 ( 1 , − 3 , 1 ) 2 11 = ( 1 11 , − 3 11 , 1 11 ) . \frac{w}{||w||}=\frac{(2,-6,2)}{\sqrt{4+36+4}}=\frac{(2,-6,2)}{\sqrt{44}}=\frac{2(1,-3,1)}{2\sqrt{11}}=(\frac{1}{\sqrt{11}},-\frac{3}{\sqrt{11}},\frac{1}{\sqrt{11}}). ∣∣ w ∣∣ w = 4 + 36 + 4 ( 2 , − 6 , 2 ) = 44 ( 2 , − 6 , 2 ) = 2 11 2 ( 1 , − 3 , 1 ) = ( 11 1 , − 11 3 , 11 1 ) .
d.) Suppose u , v u, v u , v and w w w are vectors in 3D, where u = ( u 1 , u 2 , u 3 ) , v = ( v 1 , v 2 , v 3 ) u=(u_1, u_2, u_3),\ v = (v_1, v_2, v_3) u = ( u 1 , u 2 , u 3 ) , v = ( v 1 , v 2 , v 3 ) and w = ( w 1 , w 2 , w 3 ) w=(w_1, w_2, w_3) w = ( w 1 , w 2 , w 3 ) .
( u × v ) ⋅ w = ( u 2 v 3 − v 2 u 3 , v 1 u 3 − u 1 v 3 , u 1 v 2 − v 1 u 2 ) ⋅ ( w 1 , w 2 , w 3 ) = ( u 2 v 3 − v 2 u 3 ) w 1 + ( v 1 u 3 − u 1 v 3 ) w 2 + ( u 1 v 2 − v 1 u 2 ) w 3 = ∣ u 1 u 2 u 3 v 1 v 2 v 3 w 1 w 2 w 3 ∣ . (u \times v)\cdot w=(u_2v_3-v_2u_3,v_1u_3-u_1v_3,u_1v_2-v_1u_2)\cdot(w_1,w_2,w_3)=(u_2v_3-v_2u_3)w_1+(v_1u_3-u_1v_3)w_2+(u_1v_2-v_1u_2)w_3=\begin{vmatrix}u_1 & u_ 2 & u_3 \\ v_1 & v_2 & v_3 \\ w_1 & w_2 & w_3\end{vmatrix}. ( u × v ) ⋅ w = ( u 2 v 3 − v 2 u 3 , v 1 u 3 − u 1 v 3 , u 1 v 2 − v 1 u 2 ) ⋅ ( w 1 , w 2 , w 3 ) = ( u 2 v 3 − v 2 u 3 ) w 1 + ( v 1 u 3 − u 1 v 3 ) w 2 + ( u 1 v 2 − v 1 u 2 ) w 3 = ∣ ∣ u 1 v 1 w 1 u 2 v 2 w 2 u 3 v 3 w 3 ∣ ∣ .
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