Vectors and Matrices

Vector Magnitude

Compute the Euclidean length ||v|| of a 2-component vector by squaring each component, summing the squares, and taking the square root. The example is chosen so the result is a whole number.

Example

Square the components, add the squares, and take the square root.

highlighted = computed this step

Step 1 — Set up

Start with the vector and an empty square-sum.

v=[34],∥v∥=□+□v=\begin{bmatrix}3\\4\end{bmatrix},\quad \|v\|=\sqrt{\square+\square}

Step 2 — First square

Square the first component: 3 * 3 = 9.

3⋅3=9,∥v∥=9+□3\cdot 3=9,\quad \|v\|=\sqrt{\hl{9}+\square}

Step 3 — Second square

Square the second component: 4 * 4 = 16.

4⋅4=16,∥v∥=9+164\cdot 4=16,\quad \|v\|=\sqrt{9+\hl{16}}

Step 4 — Add squares

Add the squares: 9 + 16 = 25.

9+16=25,∥v∥=259+16=\hl{25},\quad \|v\|=\sqrt{25}

Step 5 — Square root

Take the square root: sqrt(25) = 5.

25=5,∥v∥=5\sqrt{25}=\hl{5},\quad \|v\|=5

Step 6 — Result

Read the completed magnitude.

∥v∥=5\|v\|=5
Euclidean norm The magnitude (or norm) of v is ||v|| = sqrt(v[1]^2 + v[2]^2 + ...). It equals the length of the arrow from the origin to the point v.
Pythagorean triple A vector whose squared components sum to a perfect square has an integer magnitude. The 3-4-5 right triangle is the simplest example.