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=

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

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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.

Use the formula axb=

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.

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References

Tips

- 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.

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