Question

Let a parallelogram be given in space whose sides represent the vectors a and \(\mathbf{b}\). Let \(\mathbf{c}\) be a unit vector perpendicular to a plane \(C\). a) Show that \(\mathbf{a} \times \mathbf{b} \cdot \mathbf{c}\) equals plus or minus the area of the projection of the parallelogram on \(C\). b) Show that this can also be written as \(S \cos \gamma\), where \(S\) is the area of the parallelogram and \(\gamma\) is the angle between \(\mathbf{a} \times \mathbf{b}\) and \(\mathbf{c}\). c) Show that one has $$ S=\sqrt{S_{y z}^{2}+S_{z x}^{2}+S_{x y}^{2}}, $$ where \(S_{y z}, S_{z x}, S_{x y}\) are the areas of the projections of the parallelogram on the yz-plane, \(z x\)-plane, \(x y\)-plane.

Short answer

Question: Show the relationship between the area of a parallelogram formed by vectors \(\mathbf{a}\) and \(\mathbf{b}\), and the dot product of their cross product with a unit vector \(\mathbf{c}\) perpendicular to a plane C. Express this relationship in terms of area, angle, and projections, and find an expression for the area in terms of the projections onto the xy, yz, and zx planes. Answer: The area of a parallelogram with vectors \(\mathbf{a}\) and \(\mathbf{b}\) can be expressed as: \(S = \sqrt{S_{xy}^2 + S_{yz}^2 + S_{zx}^2}\) where \(S_{xy}, S_{yz}, S_{zx}\) are the areas of the projections of the parallelogram onto the xy, yz, and zx planes, respectively.

Step-by-step solution

a) Proving the relationship between cross product and projection area

First, we know that the cross product of two vectors, \(\mathbf{a}\) and \(\mathbf{b}\), produces a third vector, \(\mathbf{a} \times \mathbf{b}\), perpendicular to both \(\mathbf{a}\) and \(\mathbf{b}\). The magnitude of this cross-product vector is equal to the area of the parallelogram formed by \(\mathbf{a}\) and \(\mathbf{b}\). The dot product of this cross-product vector and the unit vector \(\mathbf{c}\), \(\mathbf{a} \times \mathbf{b} \cdot \mathbf{c}\), gives the projection of \(\mathbf{a} \times \mathbf{b}\) onto the vector \(\mathbf{c}\) in the plane \(C\). Since \(\mathbf{c}\) is a unit vector, the dot product directly calculates the magnitude of this projection, which corresponds to the area of the projected parallelogram on plane \(C\). The sign depends on the orientation of the projection with respect to the given unit vector.

b) Expressing the relationship using area and angle

Given that the area of the parallelogram is \(S\) and the angle between \(\mathbf{a} \times \mathbf{b}\) and \(\mathbf{c}\) is \(\gamma\), we can use the dot product formula to express the relationship: \(\mathbf{a} \times \mathbf{b} \cdot \mathbf{c} = |\mathbf{a} \times \mathbf{b}| |\mathbf{c}| \cos \gamma = S \cos \gamma\) Here, we have expressed the relationship mathematically, representing the same concept as before: the projection of the parallelogram's area onto plane \(C\).

c) Finding an expression for the area in terms of projections

Now, let's define \(S_{xy}, S_{yz}, S_{zx}\) as the areas of the projections of the parallelogram onto the \(xy, yz, zx\) planes, respectively. We know that: \(S_{xy} = |\mathbf{a} \times \mathbf{b}|\cdot \frac{1}{\sqrt{a_x^2 + a_y^2}} = |\mathbf{a} \times \mathbf{b}|\cos(\alpha)\) \(S_{yz} = |\mathbf{a} \times \mathbf{b}|\cdot \frac{1}{\sqrt{a_y^2 + a_z^2}} = |\mathbf{a} \times \mathbf{b}|\cos(\beta)\) \(S_{zx} = |\mathbf{a} \times \mathbf{b}|\cdot \frac{1}{\sqrt{a_z^2 + a_x^2}} = |\mathbf{a} \times \mathbf{b}|\cos(\gamma)\) Where \(\alpha, \beta, \gamma\) are the angles between the cross-product vector and the \(x, y, z\) axes respectively. Thus, we can rewrite the expressions as: \(S_{xy}^2 = |\mathbf{a} \times \mathbf{b}|^2\cos^2(\alpha)\) \(S_{yz}^2 = |\mathbf{a} \times \mathbf{b}|^2\cos^2(\beta)\) \(S_{zx}^2 = |\mathbf{a} \times \mathbf{b}|^2\cos^2(\gamma)\) Adding them together: \(S_{xy}^2 + S_{yz}^2 + S_{zx}^2 = |\mathbf{a} \times \mathbf{b}|^2(\cos^2(\alpha) + \cos^2(\beta) + \cos^2(\gamma))\) Since \(|\mathbf{a} \times \mathbf{b}|^2 = S^2\), we can simplify the expression to: \(S = \sqrt{S_{xy}^2 + S_{yz}^2 + S_{zx}^2}\) As required.

Key concepts

Cross Product

In vector mathematics, the cross product is an operation that takes two vectors and returns a third vector that is perpendicular to both of the original vectors. The direction of this new vector is defined by the right-hand rule, and its magnitude is directly related to the area of the parallelogram formed by the original vectors.

More formally, given two vectors \( \mathbf{a} \) and \( \mathbf{b} \) in three-dimensional space, their cross product \( \mathbf{a} \times \mathbf{b} \) is defined as a vector with magnitude \( |\mathbf{a} \times \mathbf{b}| \) and is given by the formula:
\[ |\mathbf{a} \times \mathbf{b}| = |\mathbf{a}||\mathbf{b}|\sin(\theta) \]
where \( \theta \) is the angle between \( \mathbf{a} \) and \( \mathbf{b} \). The result is significant in determining parallelogram area and plane projections, which are critical in multiple applications such as physics, engineering, and computer graphics.

Dot Product

The dot product or scalar product is another fundamental operation in vector algebra. When you calculate the dot product of two vectors, you get a single number, or scalar, as a result. This value measures how much one vector extends in the direction of another vector.

The formula for the dot product of two vectors \( \mathbf{a} \) and \( \mathbf{b} \) is:\[ \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}||\mathbf{b}|\cos(\phi) \]
where \( \phi \) is the angle between the vectors \( \mathbf{a} \) and \( \mathbf{b} \). This concept is crucial when dealing with vector projection, as it allows us to measure the component of one vector along the direction of another, which was utilized in the exercise to find the area of the projection of a parallelogram on a given plane.

Parallelogram Area

The area of a parallelogram can be computed using the magnitude of the cross product of its sides. If \( \mathbf{a} \) and \( \mathbf{b} \) are vectors representing the sides of a parallelogram, then its area \( S \) is given by:
\[ S = |\mathbf{a} \times \mathbf{b}| \]
The projection of a parallelogram's area onto different planes can be interpreted as finding the

Angles Between Vectors

Vectors can form angles with each other, and these angles are crucial in computing both dot and cross products. The angle between vectors is the measure of their divergence from one another and is usually denoted by Greek letters such as \( \alpha \) or \( \gamma \).

For a parallelogram formed by vectors \( \mathbf{a} \) and \( \mathbf{b} \) with the angle \( \theta \) between them, the cross product gives us a vector perpendicular to the plane containing \( \mathbf{a} \) and \( \mathbf{b} \), and the magnitude of the cross product is directly related to \( \sin(\theta) \), representing the area of the parallelogram.
Similarly, the dot product involves the angle \( \phi \) between \( \mathbf{a} \) and \( \mathbf{b} \) and the area of their projection through the \( \cos(\phi) \) factor, reflecting how parallel or aligned the vectors are.

Plane Projections

Plane projections refer to projecting a 3D shape, like a parallelogram, onto a two-dimensional plane. This is achieved by 'flattening' the figure along certain axes to obtain its shadow or outline on the plane. The area of this projection can then be determined using vector operations.

In the exercise provided, the projection of the parallelogram onto the plane \( C \) is found by the dot product, \( \mathbf{a} \times \mathbf{b} \cdot \mathbf{c} \) where \( \mathbf{c} \) is a unit vector perpendicular to plane \( C \). The result is the area of the parallelogram as viewed from the perspective of the plane onto which it's projected. Depending on \( \mathbf{c} \) direction, the projection can be positive or negative. These principles are critical in fields such as computer graphics, where objects are frequently projected onto the screen's plane.