Problem 7 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the area of the parallelogram with \(\mathbf{a}=-\mathbf{i}+\mathbf{j}-3 \mathbf{k}\) and \(\mathbf{b}=4 \mathbf{i}+2 \mathbf{j}-4 \mathbf{k}\) as the adjacent sides.

Step-by-Step Solution

Answer

The area of the parallelogram is \(2\sqrt{26}\).

Step 1: Write Vectors

Identify the vectors given in the problem. Here, the vectors are \( \mathbf{a} = -\mathbf{i} + \mathbf{j} - 3 \mathbf{k} \) and \( \mathbf{b} = 4 \mathbf{i} + 2 \mathbf{j} - 4 \mathbf{k} \).

Step 2: Compute the Cross Product

The area of the parallelogram can be computed using the magnitude of the cross product \( \mathbf{a} \times \mathbf{b} \). Calculate the cross product: \[ \mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ -1 & 1 & -3 \ 4 & 2 & -4 \end{vmatrix} \]This simplifies to: \[ \mathbf{a} \times \mathbf{b} = (1\times(-4) - (-3)\times2)\mathbf{i} - ((-1)\times(-4) - (-3)\times4)\mathbf{j} + ((-1)\times2 - 1\times4)\mathbf{k} \]which results in: \[ \mathbf{a} \times \mathbf{b} = (2)\mathbf{i} - (8)\mathbf{j} + (-6)\mathbf{k} \]

Step 3: Find the Magnitude of the Cross Product

Calculate the magnitude of the cross product vector \( \mathbf{a} \times \mathbf{b} = 2\mathbf{i} - 8\mathbf{j} - 6\mathbf{k} \). The magnitude is given by: \[ \|\mathbf{a} \times \mathbf{b}\| = \sqrt{2^2 + (-8)^2 + (-6)^2} \]Simplifying, we compute: \[ \|\mathbf{a} \times \mathbf{b}\| = \sqrt{4 + 64 + 36} = \sqrt{104} = 2\sqrt{26} \]

Step 4: Conclusion

The area of the parallelogram is equal to the magnitude of the cross product, \( \|\mathbf{a} \times \mathbf{b}\| = 2\sqrt{26} \).

Key Concepts

Cross Product

The cross product is a fundamental operation in vector calculus that is used to find a vector perpendicular to two given vectors in three-dimensional space. It has critical applications, especially in physics and engineering.

Given two vectors \( \mathbf{a} \) and \( \mathbf{b} \), the cross product \( \mathbf{a} \times \mathbf{b} \) results in a new vector, which is perpendicular to both \( \mathbf{a} \) and \( \mathbf{b} \). The main formula for computing the cross product in terms of the components of the vectors is:

• \( \mathbf{a} = a_i \mathbf{i} + a_j \mathbf{j} + a_k \mathbf{k} \)
• \( \mathbf{b} = b_i \mathbf{i} + b_j \mathbf{j} + b_k \mathbf{k} \)
• The result is \( \mathbf{c} = (a_jb_k - a_kb_j)\mathbf{i} + (a_kb_i - a_ib_k)\mathbf{j} + (a_ib_j - a_jb_i)\mathbf{k} \)

When solving the problem of finding a cross product, organize the values in a determinant matrix to ease the calculation.

In our exercise, we computed \( \mathbf{a} \times \mathbf{b} \) resulting in the vector \( 2\mathbf{i} - 8\mathbf{j} - 6\mathbf{k} \). Each component was calculated by expanding a 3x3 determinant matrix involving the unit vectors \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) and the given components of \( \mathbf{a} \) and \( \mathbf{b} \).

Parallelogram Area

The area of the parallelogram spanned by two vectors can be determined using the cross product. This is because the magnitude of the cross product gives the size of the parallelogram that the two vectors define.

The main trick to remember is:

• The vectors \( \mathbf{a} \) and \( \mathbf{b} \) represent adjacent sides of the parallelogram.
• The magnitude \( \| \mathbf{a} \times \mathbf{b} \| \) equals the area.

Why is the magnitude of the cross product equivalent to the area? It is because the cross product generates a vector whose length (magnitude) is precisely the area of the parallelogram. The direction of this vector is perpendicular to the plane containing \( \mathbf{a} \) and \( \mathbf{b} \). Hence, no component of this vector lies along the plane, ensuring the magnitude reflects only the perpendicular "height" times the base formed by \( \mathbf{a} \).

For the problem you are working on, we calculated the magnitude as \( 2\sqrt{26} \), giving us an exact measure of the parallelogram's area.

Vector Magnitude

Understanding the magnitude of a vector is crucial for numerous applications in mathematics, physics, and engineering. It represents the "size" or "length" of the vector.

Here's how you can calculate the magnitude for any vector \( \mathbf{v} = x \mathbf{i} + y \mathbf{j} + z \mathbf{k} \):

• Use the formula \( \| \mathbf{v} \| = \sqrt{x^2 + y^2 + z^2} \).
• Think of the magnitude as the distance from the origin \((0,0,0)\) to the point \((x,y,z)\) in three-dimensional space.

For our cross product vector \( \mathbf{a} \times \mathbf{b} = 2\mathbf{i} - 8\mathbf{j} - 6\mathbf{k} \), we calculated:
\[ \| \mathbf{a} \times \mathbf{b} \| = \sqrt{2^2 + (-8)^2 + (-6)^2} = \sqrt{4 + 64 + 36} = \sqrt{104} = 2\sqrt{26} \]This gives us the result we need to find the parallelogram's area, emphasizing how vector magnitude translates into practical geometrical interpretations.