How to find a vector's magnitude

A game piece can fly straight to a target three squares right and four up. Walking along the grid takes seven steps, but the flight is shorter. A vector’s magnitude means its direct length. The right triangle below links the route and the flight.

The arrow cuts across the right triangle

The dashed legs measure three and four units. The arrow is the direct length. The outer bars in the equation ask for that length.

‖A‖=Answer5

Measure the direct flight

Find the length of the move (3, 4).

Use the two short sides

$3^2+4^2=9+16=25$

The corner is a right angle, like a square’s corner. Pythagoras’ rule says the square of the long side equals the sum of the two short-side squares.
Turn the square back into a length

$\text{length}=\sqrt{25}=5$

The positive square root is the number that squares to twenty-five. Five times five is twenty-five, so the direct flight is five units.
Compare it with the grid route

$5<3+4=7$

Five measures the straight arrow. Seven measures a different route. Choose the length that answers the actual question.

Direction changes the signs, not the length

The move (−3, 4) also has length five. Squaring minus three gives nine, just as squaring three does. A length cannot be negative.

The bars around a vector ask for its magnitude. To find it, square both parts, add the squares and take the positive square root. You do not add the two parts directly.

Some lengths are not whole numbers. The move (1, 1) has length √2, about 1.41. To compare two moves, compare the sums of their squares. You can skip the roots.

Try it yourself

Work out an answer and explain your reasoning. Then check the explanation.

Find another magnitude

$\begin{bmatrix}-5\\12\end{bmatrix}$

Find the direct length. Does the minus sign change the distance?

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$\sqrt{(-5)^2+12^2}=13$

The squares add to 25 + 144 = 169. The positive square root of 169 is 13.

Separate route length from direct length

$\left\|\begin{bmatrix}6\\8\end{bmatrix}\right\|=6+8\;?$

This claim counts the two-leg route. What is the direct distance instead?

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$10$

The squared length is 36 + 64 = 100. Its positive square root is ten. Fourteen counts the route along the two legs.

Find the flight length for a new move

The game gives moves with whole-number lengths. Find the direct distance, then give one length rather than a pair.

2-minute practice · keep going after mistakes · opens in a new tab

Try vector lengths