Vectors and Matrices

Vector Addition

Add two 3-component vectors by summing corresponding components. Each component pair is handled in a single step, producing a new vector whose entries are the pairwise sums. The state panel accumulates the result one component at a time.

Example

Add matching vector components to build the sum vector one entry at a time.

highlighted = computed this step

Step 1 — Set up

Start with both vectors and an empty sum vector.

u=[3−12],v=[14−3],u+v=[□□□]u=\begin{bmatrix}3\\-1\\2\end{bmatrix},\quad v=\begin{bmatrix}1\\4\\-3\end{bmatrix},\quad u+v=\begin{bmatrix}\square\\\square\\\square\end{bmatrix}

Step 2 — First component

Add the first component: 3 + 1 = 4.

3+1=4,u+v=[4□□]3+1=4,\quad u+v=\begin{bmatrix}\hl{4}\\\square\\\square\end{bmatrix}

Step 3 — Second component

Add the second component: -1 + 4 = 3.

−1+4=3,u+v=[43□]-1+4=3,\quad u+v=\begin{bmatrix}4\\\hl{3}\\\square\end{bmatrix}

Step 4 — Third component

Add the third component: 2 + -3 = -1.

2+(−3)=−1,u+v=[43-1]2+(-3)=-1,\quad u+v=\begin{bmatrix}4\\3\\\hl{-1}\end{bmatrix}

Step 5 — Result

Read the completed sum vector.

u+v=[43−1]u+v=\begin{bmatrix}4\\3\\-1\end{bmatrix}
component-wise operation Vector addition is defined component by component: (u + v)[i] = u[i] + v[i]. The result is a vector of the same length as the inputs.