Math 150 – Fall 2015
Section 9D
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Section 9D – Vector Length
Theorem. The length of any vector hx1 , x2 i in R2 is defined to be the distance form
the origin to the point (x1 , x2 ) which is
q
||hx1 , x2 i|| = x21 + x22
We use the notation ||hx1 , x2 i|| to mean the length of the vector. A similar formula for
the length of a vector exists for vectors in Rn .
q
||hx1 , x2 , . . . , xn i|| = x21 + x22 + · · · + x2n
Example 1. Find the length of the vector h−3, 5i.
Note. We use the notation ||hx1 , x2 i|| to denote the length of a vector. This is similar
to absolute value of a real number, but we use slightly different notation to remind
ourselves that we are dealing with a vector, and not a real number.
~ and Y
~ is ||X
~ −Y
~ || (the order does not
Theorem. The distance between two vectors X
matter).
Example 2. Find the distance between the vectors h8, −4i and h−3, −2i.
Math 150 – Fall 2015
Section 9D
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~ and Y
~ be arbitrary vectors and let α and β be arbitrary scalars.
Properties: Let X
Then we have the following properties.
~ = 0 if and only if X
~ is the zero vector (the vector in which all the components
1. ||X||
are zero, i.e., h0, 0i in R2 ).
~ = |a| · ||X||.
~
2. ||αX||
That is the length of a scalar times a vector is the same as
the length of the vector times the absolute value of the scalar.
~ +Y
~ || ≤ ||X||
~ + ||Y
~ ||. This is called the triangle inequality because the vectors
3. ||X
~ Y
~ , and X
~ +Y
~ can be thought of as the three sides of a triangle, and the sum
X,
of two side of the triangle must be longer than or equal to the third side.
Example 3. Find the unit vector in the direction of h6, −7i.
Example 4. Consider a plane which is flying at 35,000 feet due north with a speed
of 450 miles per hour. Suppose our plane suddenly experiences a down draft whose
velocity is 125 mph. Find a vector to represent the planes velocity and find the speed
of the plane.
Example 5. Suppose a ferry is crossing a river 12 miles wide with a downstream
current of 4 miles per hour. The ferry goes in a direction perpendicular to the bank at
10 miles per hour. How far downstream will the ferry reach the other bank?