Area and Torque in Two Dimensions

This chapter is about the mathematical representation of some geometric and physical ideas -- including area, volume, and torque -- in our everyday two and three dimensional world. We begin in this module with area and torque in two dimensions. In the next module we look at area, volume, and torque in three dimensions.
Later in this chapter we use these idea to discuss lighting and shading of three dimensional objects,

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Suppose that x and y are vectors that represent the two sides of a parallelogram as shown in the figure at the right.
By moving the red triangle in the left hand figure below to the position indicated by in the right hand figure, we obtain a rectangle that has the same area as the original parallelogram. The length of the base of this rectangle is just ||x|| and the height of this rectangle is ||y|| sin t where t is the angle from x to y as indicated in the right hand figure below.

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Thus, the area of this rectangle and the area of the original parallelogram is

Area = ||x|| ||y|| sin t

This formula gives us a number that can be positive, negative, or zero.

It is positive if y is counterclockwise from x.
It is negative if y is clockwise from x.
It is zero if y is parallel to x.

We say that this formula gives us the signed area of the parallelogram whose sides are x and y in that order and we introduce the following terminology and notation.

Definition

Suppose that x and y are two vectors in R². Then the wedge product, x ^ y, of x and y is defined by

x ^ y = ||x|| ||y|| sin theta,

where theta is the angle from the vector x to the vector y.

If we rotate the vector y counterclockwise by pi/2 radians we get a new vector

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as shown in the figure below

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Since sin theta = - cos (theta + Pi/2) and ||y|| = ||z|| we see that

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giving us a very useful purely algebraic formula for the wedge product.

If x = (x₁, x₂) and y = (y₁, y₂) then

x ^ y = x₁ y₂ - x₂ y₁

In this module and subsequent modules we often use the notation i for the vector (1, 0) and the notation j for the vector (0, 1). Using this notation, we can write any vector (a, b) in the form

(a, b) = a i + b j

Compute each of the following using the formula above and then check your answer geometrically by drawing the appropriate parallelogram.

i ^ j answer
j ^ i answer
i ^ (a i + b j)
Prove that the wedge product has the following properties using the algebraic formula for the wedge product. Then explain each property geometrically.
x ^ y = - (y ^ x)
(cx) ^ y = c(x ^ y) = x ^ (cy) where c is any real number.
x ^ x = 0
(u + v) ^ x = (u ^ x) + (v ^ x)
x ^ (u + v) = (x ^ u) + (x ^ v)

The following theorem lists some important properties of the wedge product from the exercises above.

Theorem

x ^ y = - y ^ x
(cx) ^ y = x ^ (cy) = c(x ^ y) where c is any real number.
(u + v) ^ y = (u ^ y) + (v ^ y)
x ^ (u + v) = (x ^ u) + (x ^ v)
O ^ y = x ^ O = 0
x ^ x = 0

Sometimes it is convenient to write two two-dimensional vectors

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as the rows of a 2 by 2 matrix.

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You may have already seen the determinant, det(A), of a 2 by 2 matrix.

Definition

The determinant of a 2 by 2 matrix,

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written det(A) or

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is defined by

det(A) = a₁₁ a₂₂ - a₁₂ a₂₁.

Notice that

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so the determinant of a 2 by 2 matrix is the signed area of the parallelogram whose sides are determined by its two rows.

The two figures below each show a force represented by a vector y acting on the end of a rod represented by a vector x. One end of the rod is fixed at the origin but the rod is free to pivot about the origin.

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Because the rod is fixed at the origin the result is a "rotational force" or torque trying to rotate the rod about the origin. In the left hand picture the force will push the rod counterclockwise and in the right hand picture the force will push the rod clockwise. The resulting torque is

Torque = x ^ y

and is positive if the torque is in the counterclockwise direction and negative if it is in the clockwise direction.

Prove the statement above -- that the torque produced when a force represented by the vector y acts on a rod that is free to pivot about the origin and is represented by the vector x is
x ^ y

For each of the following, suppose that the force y is applied to the rod x. In which direction will the rod be pushed? What is the torque produced by the force on the rod?
y = i and x = 2 i + 3 j
y = i and x = 2 i - 3 j
y = 2 i + 3 j and x = i
y = i - 3 j and x = i
Suppose that the two forces (2, 4) and (3, 3) are applied to the rod (1, -1). In which direction will each individual force push the rod? In which direction will the combined force push the rod?