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

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

The distance formula for points in is analogous to the distance formula in . Given two points and , the distance between them is given by

To find the length of vector , we find the distance between and .

Distance formulas for and motivate the following definition of distance between two points in .

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 .

Practice Problems

Find the length of the following vectors. Enter your answers in exact, simplified form. (e.g. )
Answer:
Answer: Enter your answer in the most simplified exact form. (e.g. )
Answer: Enter your answer in the most simplified exact form. (e.g. )
Answer:
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:

Let . Find all possible values of if . 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.

Answer:

The points , and form a triangle in . Is it a right triangle?
Does the Pythagorean Theorem hold?