# Prove Vector Projections Perpendicular in R3

• tony873004
This is equal to v.w-(v.w)^2/v.v. But we know that v.w=|v|^2|w|cos(theta), so this is equal to |v|^2|w|cos(theta)-|v|^4|w|^2cos(theta)/(v.v). But this is equal to |v|^2|w|cos(theta)-|v|^2|w|cos(theta), which is 0. So the original equation holds and thus w-(v.w)*v/(v.v) is perpendicular to v.
tony873004
Gold Member
Let $$\overrightarrow v$$ and $$\overrightarrow w$$ be vectors in R3. Prove that $$\overrightarrow w - {\rm{proj}}_{\overrightarrow v } \overrightarrow w$$ is perpendicular to $$\overrightarrow v$$ .

Here's my attempt:
$$\begin{array}{l} \left( {\overrightarrow w - {\rm{proj}}_{\overrightarrow v } \overrightarrow w } \right) \cdot \overrightarrow v \mathop = \limits^? 0 \\ \\ {\rm{proj}}_{\overrightarrow v } \overrightarrow w = \frac{{\overrightarrow v \cdot \overrightarrow w }}{{\left| {\overrightarrow v } \right|^2 }}\overrightarrow v \\ \\ \overrightarrow v \cdot \overrightarrow w = v_1 w_1 + v_2 w_2 + v_3 w_3 \\ \\ \left| {\overrightarrow v } \right| = \sqrt {v_1^2 + v_2^2 + v_3^2 } \\ \left| {\overrightarrow v } \right|^2 = v_1^2 + v_2^2 + v_3^2 \\ \\ \frac{{\overrightarrow v \cdot \overrightarrow w }}{{\left| {\overrightarrow v } \right|^2 }} = \frac{{v_1 w_1 + v_2 w_2 + v_3 w_3 }}{{v_1^2 + v_2^2 + v_3^2 }} = \frac{{v_1 w_1 }}{{v_1^2 + v_2^2 + v_3^2 }} + \frac{{v_2 w_2 }}{{v_1^2 + v_2^2 + v_3^2 }} + \frac{{v_3 w_3 }}{{v_1^2 + v_2^2 + v_3^2 }} \\ \\ \frac{{\overrightarrow v \cdot \overrightarrow w }}{{\left| {\overrightarrow v } \right|^2 }}\overrightarrow v = \left( {\frac{{v_1 w_1 }}{{v_1^2 + v_2^2 + v_3^2 }} + \frac{{v_2 w_2 }}{{v_1^2 + v_2^2 + v_3^2 }} + \frac{{v_3 w_3 }}{{v_1^2 + v_2^2 + v_3^2 }}} \right)\left\langle {v_1 ,\,v_2 ,\,v_3 } \right\rangle \\ \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\, = \left( {\frac{{v_1 w_1 }}{{v_1^2 + v_2^2 + v_3^2 }} + \frac{{v_2 w_2 }}{{v_1^2 + v_2^2 + v_3^2 }} + \frac{{v_3 w_3 }}{{v_1^2 + v_2^2 + v_3^2 }}} \right)\left\langle {v_1 ,\,v_2 ,\,v_3 } \right\rangle \\ \end{array}$$

Things are starting to get real ugly. Am I missing an easier way?

Yes, you are. You want to show v.(w-(v.w)*v/(v.v))=0. Just multiply the outer dot product through.

## What is a vector projection?

A vector projection is a method used to find the component of a vector in a given direction. It involves finding the scalar projection, which is the length of the projection, and then multiplying it by the unit vector in the direction of the projection.

## What does it mean for two vector projections to be perpendicular?

Two vector projections are perpendicular if they form a right angle with each other. This means that their dot product is equal to zero.

## How can we prove that vector projections are perpendicular in R3?

To prove that two vector projections are perpendicular in R3, we need to show that their dot product is equal to zero. This can be done by finding the scalar projections of the two vectors and then taking their dot product. If the result is equal to zero, then the vector projections are perpendicular.

## Why is it important to understand vector projections in R3?

Understanding vector projections in R3 is important because it has many applications in physics, engineering, and mathematics. It allows us to break down complex vectors into simpler components and solve problems involving forces, motion, and geometry in three-dimensional space.

## Are there any real-world examples of vector projections in R3?

Yes, there are many real-world examples of vector projections in R3. For instance, when a plane is flying in the sky, its velocity vector can be projected onto the horizontal and vertical components to determine its speed and direction. Similarly, in construction, engineers use vector projections to determine the forces acting on a structure in different directions.

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