Length of a Vector

Vector quantities, such as velocity and force, have magnitude and direction. The magnitude of a vector quantity is the length of the vector. For example, if a force of 10 Newtons is applied to an object, we would represent the force by a 10-unit-long vector.

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The magnitude of a vector is denoted by double absolute value brackets. In the case of force \(\vec {F}\), we write

\[\norm {\vec {F}}=10\]

To find the length of a vector, we need to find the distance between the tail of the vector and its head. Recall that in \(\RR ^2\), the distance between \(A(a_1, a_2)\) and \(B(b_1, b_2)\) is given by

\begin{equation*} d_{AB}=\sqrt {(a_1-b_1)^2+(a_2-b_2)^2} \end{equation*}

A vector \(\vec {v}=\begin{bmatrix}v_1\\ v_2\end{bmatrix}\) has the length of the vector in standard position with its head at \((v_1, v_2)\) and tail at \((0, 0)\). We find the length of \(\vec {v}\) using the distance formula

\begin{equation}\label {eq:normr2} \norm {\vec {v}}=\sqrt {(v_1-0)^2+(v_2-0)^2}=\sqrt {v_1^2+v_2^2} \end{equation}

The distance formula for points in \(\RR ^3\) is analogous to the distance formula in \(\RR ^2\). Given two points \(A(a_1, a_2, a_3)\) and \(B(b_1, b_2, b_3)\), the distance between them is given by

\begin{equation*} d_{AB}=\sqrt {(a_1-b_1)^2+(a_2-b_2)^2+(a_3-b_3)^2} \end{equation*}

To find the length of vector \(\vec {v}=\begin{bmatrix}v_1\\ v_2\\ v_3\end{bmatrix}\), we find the distance between \((v_1, v_2, v_3)\) and \((0, 0, 0)\).

\begin{equation}\label {eq:normr3} \norm {\vec {v}}=\sqrt {(v_1-0)^2+(v_2-0)^2+(v_3-0)^2}=\sqrt {v_1^2+v_2^2+v_3^2} \end{equation}

Distance formulas for \(\RR ^2\) and \(\RR ^3\) motivate the following definition of distance between two points in \(\RR ^n\).

The following definition follows directly from the distance formula for \(\RR ^n\) in the same way that expressions (eq:normr2) and (eq:normr3) followed from distance formulas in \(\RR ^2\) and \(\RR ^3\).

Practice Problems

Problems -

Find the length of the following vectors. Enter your answers in exact, simplified form. (e.g. \(5\sqrt {3}\))

\[\vec {u}=\begin{bmatrix}1\\3\end{bmatrix}\]
Answer:
\[\norm {\vec {u}}=\sqrt {\answer {10}}\]
\[\vec {v}=\begin{bmatrix}-4\\-6\end{bmatrix}\]
Answer: Enter your answer in the most simplified exact form. (e.g. \(a\sqrt {b}\))
\[\norm {\vec {v}}=\answer {2}\sqrt {\answer {13}}\]
\[\vec {w}=\begin{bmatrix}2\\-4\\6\end{bmatrix}\]
Answer: Enter your answer in the most simplified exact form. (e.g. \(a\sqrt {b}\))
\[\norm {\vec {w}}=\answer {2}\sqrt {\answer {14}}\]
\[\vec {q}=\begin{bmatrix}-10\\ 4\\ -7\\ 5\\ 3\end{bmatrix}\]
Answer:
\[\norm {\vec {q}}=\sqrt {\answer {199}}\]
Find the component form of vector \(\vec {v}\) in \(\RR ^2\) if we know that \(\norm {\vec {v}}=15\), the \(x\) component of \(\vec {v}\) is \(-9\) and the vector is located in the third quadrant.

Answer:

\[\vec {v}=\begin{bmatrix}-9\\\answer {-12}\end{bmatrix}\]
For a vector in \(\RR ^2\) with a length of 26 and the \(y\) component of 10, what are the possibilities for the \(x\) component? List the possibilities in an increasing order.

Answer:

\[\answer {-24},\quad \text {and}\quad \answer {24} \]
Let \(\vec {v}=\begin{bmatrix}1\\ y\\ 4\end{bmatrix}\). Find all possible values of \(y\) if \(\norm {\vec {v}}=9\). List the possibilities in an increasing order.

Answer:

\[\answer {-8},\quad \text {and}\quad \answer {8}\]
For a vector in \(\RR ^4\), what are the possibilities for the fourth component if the length of the vector is 14, and the \(x\), \(y\) and \(z\) components are 1, 5 and 13, respectively? List the possibilities in an increasing order.

Answer:

\[\answer {-1},\quad \text {and}\quad \answer {1}\]
The points \(A(2,1,4),B(4,2,-1)\), and \(C(6,8,7)\) form a triangle in \(\RR ^3\). Is it a right triangle?
Does the Pythagorean Theorem hold?