Magnitude of a Vector

The vector is a hypotenuse

The vector a = (3, 4) goes 3 across and then 4 up. The step across is horizontal and the step up is vertical, so they meet at a right angle. The two steps and the arrow itself make a right triangle, and the arrow is its hypotenuse, the side facing the right angle.

The length of a vector is called its magnitude. The magnitude of a is written |a|, with a straight bar on each side. The same bars write the absolute value of a number, |−3| = 3, which is its size without its sign; the magnitude of a vector is its size without its direction.

The vector a = (3, 4): 3 squares across and 4 squares up from its start.

Pythagoras on the components

The legs of the right triangle are the components, 3 and 4. By Pythagoras, $|a{|}^2=3^2+4^2=9+16=25$|a|² = 3² + 4² = 9 + 16 = 25, so $|a|=\sqrt{25}=5$|a| = √25 = 5.

The same holds for any vector in two dimensions: $|\left(p,q\right)|=\sqrt{p^2+q^2}$|(p, q)| = √(p² + q²). Square each component, add the squares, and take the square root.

Most lengths are not whole numbers. The vector (2, 5) has length $\sqrt{2^2+5^2}=\sqrt{4+25}=\sqrt{29}$√(2² + 5²) = √(4 + 25) = √29, which is 5.39 to 2 decimal places.

The right triangle under a = (3, 4): legs of 3 and 4, and a hypotenuse of $\sqrt{9+16}=5$√(9 + 16) = 5, the magnitude of a.

Two sides of 3 and 4, with the angle between them as the handle. At 60° the square on the third side is 13, not 9 + 16 = 25. Turn the angle to 90°, the angle between the two components of (3, 4): only there is the square on the third side 25, and the third side, the length of the vector, 5. Pythagoras gives the magnitude because the components are perpendicular.

Negative components

A negative component makes no difference to the length, because each component is squared. $|\left(-5,12\right)|=\sqrt{{\left(-5\right)}^2+12^2}=\sqrt{25+144}=\sqrt{169}=13$|(−5, 12)| = √((−5)² + 12²) = √(25 + 144) = √169 = 13. The vectors (5, 12), (−5, 12), (5, −12) and (−5, −12) all have length 13; they differ only in direction.

The square of a negative number is positive: ${\left(-5\right)}^2=25$(−5)² = 25, not −25.

The distance between two points

To find how far apart two points are, find the vector from one to the other and then its magnitude. With A at (1, 2) and B at (7, 10), the vector from A to B is B minus A: (7 − 1, 10 − 2) = (6, 8). Its magnitude is $\sqrt{6^2+8^2}=\sqrt{36+64}=\sqrt{100}=10$√(6² + 8²) = √(36 + 64) = √100 = 10.

The vector from B to A is (−6, −8), which points the other way but has the same length, 10. A distance does not depend on which end it is measured from.

From A(1, 2) to B(7, 10) is 6 across and 8 up, so the distance AB is $\sqrt{36+64}=10$√(36 + 64) = 10.

The usual mistakes

Adding the components. For (3, 4), 3 + 4 = 7 is the distance walked across and then up; the straight arrow is shorter, 5.

Multiplying the components. 3 × 4 = 12 is twice the area of the triangle, not a length.

Stopping at the square. $3^2+4^2=25$3² + 4² = 25 is the square of the length; the length is $\sqrt{25}=5$√25 = 5.

Giving the vector instead of its length. The vector from A to B is a pair, (6, 8); its magnitude is one number, 10.

Speed is the magnitude of a velocity

A velocity is a vector: a speed in a direction. Its magnitude is the speed alone. A velocity of $\left(3,4\right)\frac{m}{s}$(3, 4) m/s, $3\frac{m}{s}$3 m/s east and $4\frac{m}{s}$4 m/s north, has magnitude $\sqrt{9+16}=5$√(9 + 16) = 5, so the speed is $5\frac{m}{s}$5 m/s.

In the application below, a boat is moved by its engine and by the current at the same time. Its velocity is the sum of the two, and the speed it really travels at is the magnitude of that sum.