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.
The magnitude of a vector is denoted by double absolute value brackets. In the case of force , we write
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 , the distance between and is given by \begin{equation*} d_{AB}=\sqrt{(a_1-b_1)^2+(a_2-b_2)^2} \end{equation*} A vector has the length of the vector in standard position with its head at and tail at . We find the length of 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 is analogous to the distance formula in . Given two points and , 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 , we find the distance between and . \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 and motivate the following definition of distance between two points in .
Let and be points in . The distance between and is defined to be \begin{equation*} d_{AB}=\sqrt{(a_1-b_1)^2+(a_2-b_2)^2+\ldots +(a_n-b_n)^2} \end{equation*}
The following definition follows directly from the distance formula for in the same way that expressions (eq:normr2) and (eq:normr3) followed from distance formulas in and .
Let be a vector in , then the length, or the magnitude, of is given by \begin{equation} \label{eq:normrn} \norm{\vec{v}}=\sqrt{v_1^2+v_2^2+\ldots +v_n^2} \end{equation}
Practice Problems
Problems prob:magnitude1-prob:magnitude4
Find the length of the following vectors. Enter your answers in exact, simplified form. (e.g. )
Find the component form of vector in if we know that , the component of is and
the vector is located in the third quadrant.
Answer:
For a vector in with a length of 26 and the component of 10, what are
the possibilities for the component? List the possibilities in an increasing
order.
Answer:
For a vector in , what are the possibilities for the fourth component if the length of
the vector is 14, and the , and components are 1, 5 and 13, respectively? List the
possibilities in an increasing order.
2024-09-11 17:59:42 Answer: