Find orthogonal given u=(1,0,-2) and v =(-3,5,1),projection of v on u and the component of vector u on v

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Jazz Frenia · Accepted Answer

We are given vectors u=(1,0,−2) and v=(−3,5,1).To find an orthogonal vector, we need to find a vector that is perpendicular to both u and v. One way to do this is to take the cross product of u and v:u×v=|i^j^k^10−2−351|=(−10,−7,5)So, the vector w=(−10,−7,5) is orthogonal to both u and v.To find the projection of v onto u, we can use the formula:projuv=v·u‖u‖2uwhere · denotes the dot product.First, we calculate ‖u‖:‖u‖=12+02+(−2)2=5Next, we calculate the dot product v·u:v·u=(−3)(1)+(5)(0)+(1)(−2)=−5Plugging these values into the formula, we get:projuv=−55(1,0,−2)=(−1,0,2)Therefore, the projection of v onto u is the vector (−1,0,2).Finally, to find the component of u on v, we can use the formula:compvu=u·v‖v‖Again, we calculate ‖v‖ first:‖v‖=(−3)2+52+12=35Next, we calculate the dot product u·v:u·v=(1)(−3)+(0)(5)+(−2)(1)=−5Plugging these values into the formula, we get:compvu=−535=−53535=−357Therefore, the component of u on v is −357.

Find orthogonal given u=(1,0,-2) and v =(-3,5,1),projection of

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