Question

Explain why u ↓( v ↓ w ) = u ↓ w.

A drawing is much appreciated

Answer 1

I really do not know what down arrow means

Answer 2

Projection

Answer 3

@sa5uk3 write out the equations

Answer 4

$$\vec{u}\downarrow\vec{w}=\frac{\vec{u}\cdot\vec{w}}{\vec{w}\cdot\vec{w}}\vec{w}\\\vec{v}\downarrow\vec{w}=\frac{\vec{v}\cdot\vec{w}}{\vec{w}\cdot\vec{w}}\vec{w}\\\vec{u}\downarrow(\vec{v}\downarrow\vec{w})=\frac{\vec{u}\cdot\frac{\vec{v}\cdot\vec{w}}{\vec{w}\cdot\vec{w}}\vec{w}}{\frac{\vec{v}\cdot\vec{w}}{\vec{w}\cdot\vec{w}}\vec{w}\cdot\frac{\vec{v}\cdot\vec{w}}{\vec{w}\cdot\vec{w}}\vec{w}}\frac{\vec{v}\cdot\vec{w}}{\vec{w}\cdot\vec{w}}\vec{w}=\frac{\vec{v}\cdot\vec{w}}{\vec{w}\cdot\vec{w}}\frac{\vec{w}\cdot\vec{w}}{\vec{v}\cdot\vec{w}}\frac{\vec{w}\cdot\vec{w}}{\vec{v}\cdot\vec{w}}\frac{\vec{v}\cdot\vec{w}}{\vec{w}\cdot\vec{w}}\frac{\vec{u}\cdot\vec{w}}{\vec{w}\cdot\vec{w}}\vec{w}\\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =\frac{\vec{u}\cdot\vec{w}}{\vec{w}\cdot\vec{w}}\vec{w}=\vec{u}\downarrow\vec{w}$$

Answer 5

I must prove it via drawing. Thanks for your explanation though.

Answer 6

your initial statement is not always true

Answer 7

you should add that \(\vec{v}\) and \(\vec{w}\) are not orthogonal.

Answer 8

I need to show when it is true. In terms of a drawing