Question

If v=2i+5j and w=8i+20j, find proj jw​v. Then decompose v into two vectors v1​ and v2​, where v1​ is parallel to w and v2​ is orthogonal to w prolw​v= (Simplify your answer. Use integers or fractions for any numbers in the expression. Type your answer in terms of i and ji)

Answer 1

Vector v projected onto w results in projwv, which is a vector parallel to w. Since v and w are parallel, the orthogonal component is zero and v itself is the decomposition parallel to w.

Step-by-step explanation:

The projection of vector v onto vector w, denoted as projwv, is found by taking the dot product of v with a unit vector in the direction of w and scaling it by the magnitude of w. Since w = 8i + 20j is parallel to v = 2i + 5j, we only need to consider the magnitudes and directions in the projection calculation. The resulting projection, projwv, is a vector parallel to w and the orthogonal component, v2, is the zero vector since v is already parallel to w.

The decomposition of v requires finding vectors v1 and v2 such that v1 is parallel to w and v2 is orthogonal to w. In this case, v1 would be the same as v itself and v2 would be zero.