# Find quantity of vector projection

• MHB
• evinda
Thinking)Yeah, you got it! Great job! (Smile)In summary, we discussed the projection of a vector onto a subspace and how to find the coordinates of the projection using the vector rejection formula. We also learned the vector projection formula and how to use it when the vectors span the subspace instead of being perpendicular to it. Finally, we solved a problem involving finding the number $2x+7y+3z$ when given a projection onto a subspace.
evinda
Gold Member
MHB
Hello! (Wave)

Let $W$ be the subspace of $\mathbb{R}^3$ that is orthogonal to the vector $w_1=(-1,-1,1)$ and $p=(x,y,z)$ the projection of the vector $v=(-1,1,2)$ onto $W$. What is $7x-11y+5z$ equal to?I have thought the following:

$\text{proj}_Wv=\frac{\langle v, w_1\rangle}{\langle w_1, w_1\rangle} w_1 \Rightarrow (x,y,z)=\frac{(-1,1,2) \cdot (-1,-1,1)}{(-1,-1,1) \cdot (-1,-1,1)} (-1,-1,1) =\frac{1-1+2}{1+1+1}(-1,-1,1)=\frac{2}{3}(-1,-1,1)=\left( -\frac{2}{3},-\frac{2}{3},\frac{2}{3}\right)$

Then we get that

$$7x-11y+5z=7 \left( -\frac{2}{3}\right)-11\left( -\frac{2}{3}\right)+5\left( \frac{2}{3}\right)=6$$Have I done something wrong? Because the possible answers are $-4,-12,9,15,-14,13$.

evinda said:
Hello! (Wave)

Let $W$ be the subspace of $\mathbb{R}^3$ that is orthogonal to the vector $w_1=(-1,-1,1)$ and $p=(x,y,z)$ the projection of the vector $v=(-1,1,2)$ onto $W$. What is $7x-11y+5z$ equal to?I have thought the following:

$\text{proj}_Wv=\frac{\langle v, w_1\rangle}{\langle w_1, w_1\rangle} w_1 \Rightarrow (x,y,z)=\frac{(-1,1,2) \cdot (-1,-1,1)}{(-1,-1,1) \cdot (-1,-1,1)} (-1,-1,1) =\frac{1-1+2}{1+1+1}(-1,-1,1)=\frac{2}{3}(-1,-1,1)=\left( -\frac{2}{3},-\frac{2}{3},\frac{2}{3}\right)$

Then we get that

$$7x-11y+5z=7 \left( -\frac{2}{3}\right)-11\left( -\frac{2}{3}\right)+5\left( \frac{2}{3}\right)=6$$Have I done something wrong? Because the possible answers are $-4,-12,9,15,-14,13$.

Hey evinda!

Your projection on $W$ is actually the projection on $W^\perp$.
$$\text{proj}_Wv=v-\text{proj}_{W^\perp}v=v - \frac{\langle v, w_1\rangle}{\langle w_1, w_1\rangle} w_1$$
(Thinking)

I like Serena said:
Hey evinda!

Your projection on $W$ is actually the projection on $W^\perp$.
$$\text{proj}_Wv=v-\text{proj}_{W^\perp}v=v - \frac{\langle v, w_1\rangle}{\langle w_1, w_1\rangle} w_1$$
(Thinking)

Ah! Is it known that this formula represents the projection of $v$ onto $W$ ? (Thinking)

Using this formula, I got that $(x,y,z)=\left( -\frac{1}{3}, \frac{5}{3}, \frac{4}{3}\right)$ and thus $7x-11y+5z=-14$.

evinda said:
Ah! Is it known that this formula represents the projection of $v$ onto $W$ ? (Thinking)

Using this formula, I got that $(x,y,z)=\left( -\frac{1}{3}, \frac{5}{3}, \frac{4}{3}\right)$ and thus $7x-11y+5z=-14$.

It's indeed a known formula and it's called vector rejection.
We can see why it works in this picture:
\begin{tikzpicture}[>=stealth',rotate=10]
%preamble \usetikzlibrary{arrows}
\draw[dashed] (3,0) -- (3,4);
\draw[dashed] (0,-1) -- (0,5) node
{$w_1^\perp$};
\draw[->] (0,0) -- (5,0) node[above] {$w_1$};
\draw[ultra thick,blue,->] (0,0) -- (3,4) node[above] {$v$};
\draw[->] (3,4) -- (0,4);
\draw[thick,->] (0,0) -- node
{$v-\frac{v\cdot w_1}{w_1\cdot w_1}w_1$} (0,4);
\draw[thick,->] (0,0) -- node[below] {$\frac{v\cdot w_1}{w_1\cdot w_1}w_1$} (3,0);
\end{tikzpicture}

The vector $\frac{v\cdot w_1}{w_1\cdot w_1}w_1$ is the so called vector projection.
From the picture we can see how we can construct the vector projection on the perpendicular space.
And if $w_1$ is a vector with length 1, these formulas simplify to $(v\cdot w_1)w_1$ respectively $v-(v\cdot w_1)w_1$. (Nerd)​

I like Serena said:
It's indeed a known formula and it's called vector rejection.
We can see why it works in this picture:
\begin{tikzpicture}[>=stealth']
%preamble \usetikzlibrary{arrows}
\draw[dashed] (3,0) -- (3,4);
\draw[dashed] (0,-1) -- (0,5) node
{$w_1^\perp$};
\draw[->] (0,0) -- (5,0) node[above] {$w_1$};
\draw[ultra thick,blue,->] (0,0) -- (3,4) node[above] {$v$};
\draw[->] (3,4) -- (0,4);
\draw[thick,->] (0,0) -- node
{$v-\frac{v\cdot w_1}{w_1\cdot w_1}w_1$} (0,4);
\draw[thick,->] (0,0) -- node[below] {$\frac{v\cdot w_1}{w_1\cdot w_1}w_1$} (3,0);
\end{tikzpicture}

The vector $\frac{v\cdot w_1}{w_1\cdot w_1}w_1$ is the so called vector projection.
From the picture we can see how we can construct the vector projection on the perpendicular space.
And if $w_1$ is a vector with length 1, these formulas simplify to $(v\cdot w_1)w_1$ respectively $v-(v\cdot w_1)w_1$. (Nerd)​

Ok, thanks a lot! (Smirk)​

With the same logic, I wanted to solve the following:

Let $W$ the subspace of $\mathbb{R}^3$ that is spanned by $w_1=(1,-1,0)$, $w_2=(1,1,2)$ and $p=(x,y,z)$ the projection of the vector $v=(-1,1,2)$ onto $W$. What is the number $2x+7y+3z$ equal to?

Using the formula, I got that

$$(x,y,z)=(-1,1,2)- \left( \frac{(-1,1,2) \cdot (1,-1,0)}{(1,-1,0) \cdot (1,-1,0)}(1,-1,0)+\frac{(-1,1,2) \cdot (1,1,2)}{(1,1,2) \cdot (1,1,2)}(1,1,2)\right)=\left( -\frac{2}{3},-\frac{2}{3}, \frac{2}{3}\right)$$

Then $2x+7y+3z=-4$. But this isn't a possible answer... (Worried)

Have I done something wrong? (Thinking)

evinda said:
With the same logic, I wanted to solve the following:

Let $W$ the subspace of $\mathbb{R}^3$ that is spanned by $w_1=(1,-1,0)$, $w_2=(1,1,2)$ and $p=(x,y,z)$ the projection of the vector $v=(-1,1,2)$ onto $W$. What is the number $2x+7y+3z$ equal to?

Using the formula, I got that

$$(x,y,z)=(-1,1,2)- \left( \frac{(-1,1,2) \cdot (1,-1,0)}{(1,-1,0) \cdot (1,-1,0)}(1,-1,0)+\frac{(-1,1,2) \cdot (1,1,2)}{(1,1,2) \cdot (1,1,2)}(1,1,2)\right)=\left( -\frac{2}{3},-\frac{2}{3}, \frac{2}{3}\right)$$

Then $2x+7y+3z=-4$. But this isn't a possible answer... (Worried)

Have I done something wrong? (Thinking)

This time round we have vectors that span $W$ instead of being perpendicular to $W$.
So we need the vector projection formula instead of the vector rejection formula.

Alternatively we can find the vector $n$ that is perpendicular to both $w_1$ and $w_2$, and then we can use the vector rejection formula. (Thinking)

I like Serena said:
This time round we have vectors that span $W$ instead of being perpendicular to $W$.
So we need the vector projection formula instead of the vector rejection formula.

Alternatively we can find the vector $n$ that is perpendicular to both $w_1$ and $w_2$, and then we can use the vector rejection formula. (Thinking)

Ah ok... So we have that $(x,y,z)=\frac{(-1,1,2) \cdot (1,-1,0)}{(1,-1,0) \cdot (1,-1,0)}(1,-1,0)+\frac{(-1,1,2) \cdot (1,1,2)}{(1,1,2) \cdot (1,1,2)} \cdot (1,1,2)=\left( -\frac{1}{3}, \frac{5}{3}, \frac{4}{3}\right)$ and thus $2x+7y+3z=15$, right? (Thinking)

evinda said:
Ah ok... So we have that $(x,y,z)=\frac{(-1,1,2) \cdot (1,-1,0)}{(1,-1,0) \cdot (1,-1,0)}(1,-1,0)+\frac{(-1,1,2) \cdot (1,1,2)}{(1,1,2) \cdot (1,1,2)} \cdot (1,1,2)=\left( -\frac{1}{3}, \frac{5}{3}, \frac{4}{3}\right)$ and thus $2x+7y+3z=15$, right? (Thinking)

Yep. (Nod)

I like Serena said:
Yep. (Nod)

Nice... thank you (Smirk)

## What is vector projection?

Vector projection is a mathematical operation that involves finding the component of one vector in the direction of another vector. It is used to determine the amount of one vector that "projects" onto another vector.

## How do you calculate the quantity of vector projection?

The quantity of vector projection can be calculated by taking the dot product of the two vectors and dividing it by the magnitude of the second vector. This can also be represented as the length of the first vector multiplied by the cosine of the angle between the two vectors.

## What is the significance of vector projection in science?

Vector projection is used in various fields of science, such as physics and engineering, to solve problems involving forces, motion, and vectors. It is also used in computer graphics, where it is used to determine the position of objects in 3D space.

## Can vector projection be negative?

Yes, vector projection can be negative. This indicates that the two vectors are pointing in opposite directions, and the quantity of projection is in the opposite direction of the vector being projected onto.

## In what scenarios would you use vector projection?

Vector projection is commonly used in physics to calculate the work done by a force, in mechanics to determine the displacement of an object, and in computer science for graphics rendering and animation. It is also used in real-world applications such as navigation and satellite positioning systems.

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