Perpendicular Vectors Section 8-4

WHAT YOU WILL LEARN: 1.How to find the inner product and cross product of two vectors. 2.How to determine whether two vectors are perpendicular.

Inner Product of Vectors The inner product of vectors a plane: If a and b are two vectors and,the inner Product of a and b is defined as: This is read as “a” dot “b” and is called the dot product It takes two vector quantities and produces a single “number” value (one vector “superimposed” on another). Two vectors that are perpendicular to one another will have an inner product of 0.

Perpendicular Vectors The “ ratio ” of horizontal and vertical components can be thought of as the slopes of the lines they lie on. When two vectors are perpendicular, their slopes are opposite reciprocals. Example: Find each inner product if p=, q= and m=. Are any of the pair of vectors perpendicular?

You Try Find each inner product if x=, y=, z=. Are any pairs perpendicular?

Perpendicularity in 3 Dimensions Inner product of vectors in space: If a = and b = then Example: Find the inner product of a and b if: a= and b=. Are they perpendicular?

Cross Product This is function that allows us to kind of “ multiply ” two vectors and get a new vector. The resulting vector does not lie in the lane of the given vectors, but is perpendicular to the plane containing the two vectors. The cross product of a and b is written as a x b. Cross product of vectors in space: If a = and b =, then the cross product of a and b is defined as follows:

An Example Find the cross product of v and w if v = and w =. Verify that the resulting vector is perpendicular to v and w.

You Try Find the cross product of a and b if a = and b =. Verify that the resulting vector is perpendicular to a and b.

How do you use this stuff? Glad you asked. In physics, the torque T about a point A created by a force F at a point B is given by T = AB x F. The magnitude of T represents the torque in foot-pounds.

A Word Problem Suppose a race car driver is applying a force of 25 pounds along the positive z-axis to the gearshift of his car. If the center of the connection of the gearshift is at the origin, the force is applied at the point (0.75, 0, 0.27). Find the torque. Use T = AB x F Step 1: Find AB Step 2: F is (force along z axis) Step 3: Find T Step 4: Find the magnitude of T

You Try A monster truck driver is applying a force of 28 pounds along the positive z-axis to the gearshift of his truck. If the center of the gearshift connection is at the point x, the force is applied at the point (0.70, 0, 0.31). Find the torque.

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