How to Multiply Vectors

By Nicole Newman

A vector is defined as a quantity with both direction and magnitude. Two vectors can be multiplied to yield a scalar product through the dot product formula. The dot product is used to determine if two vectors are perpendicular to one another. On the other hand, two vectors can produce a third, resultant vector using the cross product formula. The cross product arranges the vector components in a matrix of rows and columns. It allows the student to determine the resultant force's magnitude and direction with little effort.

The Dot Product

Compute the dot product for two given vectors a=<a1, a2, a3> and b=<b1, b2, b3> to obtain the scalar product, (a1_b1)+(a2_b2)+(a3*b3).

Compute the dot product for the vectors a=<0,3,-7> and b=<2, 3, 1> and obtain the scalar product, which is 0(2)+3(3)+(-7)(1), or 2.

Find the dot product of two vectors if you are given the magnitudes and angle between the two vectors. Determine the scalar product of a=8, b=4 and theta=45 degrees using the formula |a| |b| cos theta. Obtain the final value of |8| |4| cos (45), or 16.81.

The Cross Product

Use the formula axb=<a2b3-a3b2, a3b1-a1b3, a1b2-a2b1> to determine the cross product of vectors a and b.

Find the cross products of vectors a=<2, 1, -1> and b=<-3,4,1>. Multiply vectors a and b using the cross product formula to obtain <(1_1)-(-1_4), (-1_-3)-(2_1), (2_4)-(1_-3)>.

Simplify your response to <1+4, 3-2, 8+3>, or <5, 1, 11>.

Write your answer in the i, j, k component form by converting <5. 1. 11> to 5i+j+11k.

Tip

If axb=0, then the two vectors are parallel to one another. If the multiplied vectors do not equal zero, then they are perpendicular vectors.

About the Author

Nicole Newman is a Dartmouth College associate who works in Tiltfactor Laboratory, Dartmouth's premier game design center. Her research has included investigating the digital humanities through "Writing as a Dimensional Artifact" and "Evolution of the Ghetto: The Decline of America’s Inner Cities," a research initiative on urban design.