Definition: The magnitude of a vector is its length, found with the Pythagorean theorem: for the vector with components a and b, the magnitude is the square root of a squared plus b squared.
What is magnitude?
The magnitude of v = <a, b> is written |v| and equals the square root of (a2 + b2). In three dimensions, add c2 under the root. It is always zero or positive.
How do you also get direction?
The direction angle theta satisfies tan theta = b/a, but check the quadrant of the vector.
Worked example: v = <-7, 24>. Then |v| = square root of (49 + 576) = square root of 625 = 25.
The reference angle is the inverse tangent of 24/7, about 73.74 degrees. The vector points into quadrant II, so the direction is 180 – 73.74 = 106.26 degrees.
What mistake do students make?
Using the calculator’s inverse tangent without checking the quadrant, which gives -73.74 degrees and points the wrong way. Mathbot sketches the vector before trusting the angle.