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…
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…