Operations with Vectors

Operations with Vectors

Addition

Given $\vec{v}= \langle \color{#8925AE}{v_1},\color{#8925AE}{v_2} \rangle$ and let $\vec{u}= \langle \color{#29A603}{u_1},\color{#29A603}{u_2} \rangle$.

Then, $\vec{v}+\vec{u}= \langle \color{#8925AE}{v_1}+\color{#29A603}{u_1},\color{#8925AE}{v_2}+\color{#29A603}{u_2} \rangle$

Subtraction

Given $\vec{v}= \langle \color{#8925AE}{v_1},\color{#8925AE}{v_2} \rangle$ and let $\vec{u}=\langle \color{#29A603}{u_1},\color{#29A603}{u_2} \rangle$.

Then, $\vec{v}-\vec{u}= \langle \color{#8925AE}{v_1}-\color{#29A603}{u_1},\color{#8925AE}{v_2}-\color{#29A603}{u_2} \rangle$

Scalar Multiplication

Given $\vec{v}= \langle \color{#8925AE}{v_1},\color{#8925AE}{v_2} \rangle$ and scalar $\color{green}{c}$.

Then, $c\vec{v}= \langle \color{#29A603}{c}\color{#8925AE}{v_1},\color{#29A603}{c}\color{#8925AE}{v_2} \rangle$

Magnitude

Given $\vec{v}= \langle \color{#8925AE}{v_1},\color{#8925AE}{v_2} \rangle$.

Then the magnitude of $\vec{v}$, or $\Vert \vec{v} \Vert$, is $\sqrt{\color{#8925AE}{v_1}^2+\color{#8925AE}{v_2}^2}$.

(Video) Vector Operations

by Math and Stats Help

This video by Math and Stats Help reviews the basic operations that we can do with vectors.

Operations on Vectors

Given $u= <2, 3>$ and $v=<-1, 4>$, find:

A) $2u$

B) $2u+v$

C) $v-3u$

Be mindful of the signs for the vector quantities.

We're going to look at A first. This is called scalar multiplication because we're looking at a scaled factor of $2$. So, we're taking the vector, $\vec{u}$ and multiplying it by $2$.

$$2 \cdot \vec{u} = 2 <2,3>$$

We'll sort of use the distributive property to distribute the scaled factor of $2$.

$$2 \cdot \vec{u} = 2 <2,3> = <4,6>$$

Now, let's look at B. Here, we will be adding, in addition to the scalar multiplication we saw earlier.

$$2 \cdot \vec{u} + \vec{v} = <4,6> + <-1,4>$$

To add the two vectors together, you need to add corresponding elements.

$$<4,6> + <-1,4> = <3, 10>$$

Lastly, we will look at C. Here we see vector subtraction and scalar multiplication.

$$\vec{v} - 3\vec{u} = <-1,4>-3<2, 3>$$

We still want to be aware of PEMDAS, so we're going to do the scalar multiplication first.

$$<-1,4>-3<2, 3> = <-1, 4> + <-6, -9> = <-7, -5>$$

These are our resultant vectors:

A) $2u = <4,6>$

B) $2u+v = <3, 10>$

C) $v-3u = <-7, -5>$

Operations with Vectors

There are four main operations that we can do with vectors.

Looking Further

Vector-valued functions provide another way of parametrizing a curve by expressing position as a vector that changes with a parameter, often time. Instead of tracking each coordinate separately, the vector form captures both components together and makes operations like addition and scalar multiplication more intuitive. This perspective connects vector operations directly to motion and geometry, setting the stage for deeper work with parametric curves and vector functions.

Other Topics with Vectors

Get ready to be a master of vectors! You can learn about the dot products, magnitude, the cross product, and vector projections.