Let u, v, and w be arbitrary vectors in 3–space. Using the geometric principles
ID: 2892411 • Letter: L
Question
Let u, v, and w be arbitrary vectors in 3–space. Using the geometric principles of the dot product and cross product,
(a) describe u as the sum of two vectors, of which one is parallel to v and the other is orthogonal to v,
(b) find a vector that is orthogonal to both u and v,
(c) find a vector that is orthogonal to each of u, v, and w,
(d) find a scalar that provides the volume of the parallelepiped induced by u, v, and w,
e) find a vector that is orthogonal to u × v and u × w,
(f) find a vector of length |u| in the direction of v.
THis is more of a abstract type question so please instead of just the answers can you please explain along the way too just so i know whats going on instead of just looking at the answers!
Thank you!
Explanation / Answer
(a) In this part we need to express vector u as sum of two vectors one of which is parallel to vector v and another one is perpendicular to v.
This can done by finding projections of u along v and perpendicular to v and then summing those up.
Projection of u along v will be (u.v)v/|v|^2 and projection of u perpendicular to v will be u-(u.v)v/|v|^2
on addving both these components, we will get the original vector u.
(b) We know that if we have two vector u and v and we have to find a vector which is orthogonal to both u and v, we simply do the cross product of the two vectors because crossproduct is always perpendicular to the plan containing original vectors. Therefore, the required vector is uxv
(c) This can be ontained as (uxv)x(uxw)
(d) Scalar that provides the volume of a parallelopiped with sides along u, v and w can always be found by taking scalar triple product of the vectors. Therefore, the required volume is u.(vxw)
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