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Two non-zero vectors are said to be perpendicular or orthogonal if the included angle
is a right angle.
Let and be non-zero vectors in . Then if and only if and are orthogonal.
Recall that to prove an “if and only if” statement, we must prove two
statements. In this particular case, we will need to show the following
If , then and are orthogonal.
If and are orthogonal, then .
We leave the details of the proof to the reader.
Find the degree measure of the included angle, for each pair of vectors. Round your
answers to the nearest tenth.
What does the sign of the dot product tell us about the included angle?
Find all values of so that is orthogonal to . List your answers in increasing
Find the value of for which the vector is parallel to the vector . What is the
measure of the included angle, ? Find the measure of the included angle using
Theorem ex:anglebetweenvectors. Do the two results agree?
Prove that if is a unit vector, then .
Prove that if and are unit vectors, then . In what cases are the extreme values of 1
Imagine a clock with hands represented by vectors and , as shown below. At what
whole hour will attain its maximum value? At what whole hour will be as small as
Let be a circle of radius . Suppose and are the endpoints of a diameter
of , and is a point on distinct from and . Show that vectors and are
Assign coordinates to points , and , express vectors and in component form, then
find the dot product of and .
A rhombus is a quadrilateral with four congruent sides. Use vectors to prove that a
parallelogram is a rhombus if and only if its diagonals are perpendicular.
on vector subtraction in Module VEC-M-0030.