[From: ] [author: ] [Date: 11-04-24] [Hit: ]
v1 = (v∙w)/||w|| [w/||w||] = (v∙w/||w||²) w.Note that v∙w/||w|| is a scalar, and w/||w|| is a unit vector parallel to w.This is very useful when dealing with forces later when dealing with surfaces and solids in space.http://www.vitutor.......
i learned vector and know how to express cosine by vector,but what is scalar projection and how do you prove that and what is it use for?

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People use various terms here, but I think you are talking about the scalar component of one vector in the direction of (onto) another. If v and w are non-parallel vectors, and w is not the zero vector, then we can write v as a sum

v = v1 + v2

where v1 is parallel to w and v2 is perpendicular to w. The vector v1 is called the vector projection of v onto w. The scalar projection is the magnitude of v1.

The vector projection formula is

v1 = (v∙w)/||w|| [w/||w||] = (v∙w/||w||²) w.

Note that v∙w/||w|| is a scalar, and w/||w|| is a unit vector parallel to w. The scalar projection is

v∙w/||w|| <----that is v dotted with w divided by magnitude of w.

This is very useful when dealing with forces later when dealing with surfaces and solids in space.

http://www.vitutor.com/geometry/vec/vect…
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