Cambridge O Level Mathematics (Syllabus D) · Syllabus 4024 · Transformations and Vectors
Magnitude of a Vector
What is Magnitude of a Vector?
The magnitude of a vector is its length, written between modulus signs, so the magnitude of a is written mod a and the magnitude of the directed segment AB is written mod AB. For a column vector with entries x and y the magnitude is the square root of x squared plus y squared, because the two components are the perpendicular legs of a right-angled triangle whose hypotenuse is the vector itself, so the result is Pythagoras theorem applied to a displacement. A magnitude is a scalar, not a vector: it is a single non-negative number that measures how long the vector is and says nothing about which way it points. Squaring removes any minus signs, so a vector and its reverse always have the same magnitude, and the answer can never come out negative. Magnitudes may be left exact as an integer or a surd, or given as a decimal to three significant figures when a decimal is asked for. Multiplying a vector by a scalar k multiplies its magnitude by the modulus of k. The magnitude formula is not printed in the list of formulas on the examination paper, so it has to be recalled.
This definition is part of the Transformations and Vectors chapter in Cambridge O Level Mathematics (Syllabus D).
Questions students ask about Magnitude of a Vector
How do you find the magnitude of a vector?
By Pythagoras, because the two components are perpendicular: \(\left|\begin{pmatrix}x\y\end{pmatrix}\right|=\sqrt{x^{2}+y^{2}}\). For \(\begin{pmatrix}-6\8\end{pmatrix}\) this is \(\sqrt{36+64}=10\). A magnitude is a non-negative scalar, never a column, so a negative answer is a signal to check. Leave it as an exact surd unless a decimal is asked for, in which case give 3 significant figures.