AI Tutor
If a drone flies 3 miles east and 4 miles north, how far is it from its launchpad in a straight line?
The straight-line length of any vector is called its magnitude or Euclidean norm, written with double bars as βˆ₯vβˆ₯\|\mathbf{v}\|.

Magnitude in Two Dimensions

Because perpendicular xx and yy components meet at a right angle, they form the legs of a right triangle where the vector itself is the hypotenuse.
Over 2,500 years ago, Greek mathematicians codified the rule that the hypotenuse squared equals the sum of the squared legs, giving us the formula βˆ₯vβˆ₯=vx2+vy2\|\mathbf{v}\| = \sqrt{v_x^2 + v_y^2}.
Let's calculate the magnitude of the vector a=βŸ¨βˆ’6,8⟩\mathbf{a} = \langle -6, 8 \rangle.
βˆ₯aβˆ₯=(βˆ’6)2+82=36+64=100=10\|\mathbf{a}\| = \sqrt{(-6)^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10
A common trap is writing βˆ’62=βˆ’36-6^2 = -36; remember that squaring any real negative component always produces a positive value: (βˆ’6)Γ—(βˆ’6)=+36(-6) \times (-6) = +36.
What happens when our drone climbs into the sky, adding an altitude axis to its journey?

Magnitude in Three Dimensions

In 3D space, a vector has three perpendicular components: v=⟨vx,vy,vz⟩\mathbf{v} = \langle v_x, v_y, v_z \rangle.
To find its length, we apply the Pythagorean theorem twice: first on the flat ground to find base diagonal d=vx2+vy2d = \sqrt{v_x^2 + v_y^2}, and then vertically with height vzv_z to get βˆ₯vβˆ₯=d2+vz2=vx2+vy2+vz2\|\mathbf{v}\| = \sqrt{d^2 + v_z^2} = \sqrt{v_x^2 + v_y^2 + v_z^2}.