Chapter 13. Section 13.3 The Dot Product. Dot Product The dot product of two vectors u and v is a number (a scalar) that can be computed in the following ways: Geometrically the dot product gives information about the angle between the vectors u and v , specifically:. u. . v.
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The Dot Product
The dot product of two vectors u and v is a number (a scalar) that can be computed in the following ways:
Geometrically the dot product gives information about the angle between the vectors u and v, specifically:
If u and v are nonzero vectors the only way that is to have which means that (or ). This means the vectors are perpendicular which we call orthogonal. The vector u is orthogonal to vector v if and only if .
Algebraic Properties of Dot Product
Let u, v, and w be vectors and r a scalar.
and if and only if (Here is the zero vector.)
If u and v are nonzero vectors if and only if
Find the angle between the vectors u and v given as:
The projection of a vector u onto a non zero vector v is a vector parallel to v whose difference with u is orthogonal to v.
To derive a formula for this let h be the length that v must be rescaled to get an orthogonal vector.
Multiply the unit vector in v’s direction by h to get the projection.
Find all values for c so that the vectors u and v given to the right are perpendicular.
The idea is to find the dot product of u and v and set it equal to zero.
Setting equal to zero and solving:
We get the solutions and
Show that for any two nonzero vectors u and v the two vectors v and are perpendicular vectors.
The formula for
Now take the dot product of v and and simplify it.