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.