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Chapter 2: Problem 202

Find vector \(\mathbf{c} \times(\mathbf{a}+3 \mathbf{b}),\) where \(\mathbf{a}=\left|\begin{array}{ccc}\mathbf{i} & \mathbf{j} & \mathbf{k} \\ 5 & 0 & 9 \\ 0 & 1 & 0\end{array}\right|\), \(\mathbf{b}=\left|\begin{array}{rrr}\mathbf{i} & \mathbf{j} & \mathbf{k} \\ 0 & -1 & 1 \\ 7 & 1 & -1\end{array}\right|, \quad\) and \(\mathbf{c}=\mathbf{i}-\mathbf{k} .\)

Short Answer

Expert verified
The vector is \((-21, -20, -21)\).

Step by step solution

01

Understand the Problem

We need to find the cross product \( \mathbf{c} \times (\mathbf{a} + 3\mathbf{b}) \). First, we need to determine the vectors \( \mathbf{a}, \mathbf{b}, \) and \( \mathbf{c} \). Given the determinants for \( \mathbf{a} \) and \( \mathbf{b} \), we must calculate these vectors as well as their sum after multiplying \( \mathbf{b} \) by 3.
02

Calculate the Vector \( \mathbf{a} \)

To find \( \mathbf{a} \), compute the determinant of the matrix associated with it. The determinant will give us the vector form. This matrix is: \[\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ 5 & 0 & 9 \ 0 & 1 & 0 \end{vmatrix}\]Expanding along the first row, we find:\[\mathbf{a} = (0 \cdot 0 - 9 \cdot 1) \mathbf{i} - (5 \cdot 0 - 0 \cdot 0) \mathbf{j} + (5 \cdot 1 - 0 \cdot 0) \mathbf{k} = -9\mathbf{i} + 5\mathbf{k}\]So, \( \mathbf{a} = (-9, 0, 5) \).
03

Calculate the Vector \( \mathbf{b} \)

Similarly, find \( \mathbf{b} \) from its determinant:\[\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ 0 & -1 & 1 \ 7 & 1 & -1 \end{vmatrix}\]Expanding along the first row, we get:\[\mathbf{b} = ((-1)(-1) - 1 \cdot 1) \mathbf{i} - (0)(-1) - (7 \cdot 1) \mathbf{j} + (0 \cdot 1 - (-1)(7)) \mathbf{k}\]\[= 1\mathbf{i} - 7\mathbf{j} + 7\mathbf{k}\]Thus, \( \mathbf{b} = (1, -7, 7) \).
04

Compute \( 3\mathbf{b} \)

Compute \( 3\mathbf{b} \) by multiplying each component of \( \mathbf{b} \) by 3:\[3\mathbf{b} = 3(1, -7, 7) = (3, -21, 21)\].
05

Find Vector \( \mathbf{a} + 3\mathbf{b} \)

Add the vectors \( \mathbf{a} = (-9, 0, 5) \) and \( 3\mathbf{b} = (3, -21, 21) \):\[( -9 + 3, 0 - 21, 5 + 21) = (-6, -21, 26)\]Thus, \( \mathbf{a} + 3\mathbf{b} = (-6, -21, 26) \).
06

Find the Cross Product \( \mathbf{c} \times (\mathbf{a} + 3\mathbf{b}) \)

Calculate the cross product of \( \mathbf{c} = (1, 0, -1) \) with \( \mathbf{a} + 3\mathbf{b} = (-6, -21, 26) \):Use the formula for cross product:\[\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ 1 & 0 & -1 \ -6 & -21 & 26 \end{vmatrix}\]This expands to:\[(0 \cdot 26 - (-21) (-1))\mathbf{i} - (1 \cdot 26 - (-1)(-6))\mathbf{j} + (1 \cdot (-21) - 0(-6))\mathbf{k}\]\[= (-21)\mathbf{i} - (26 - 6)\mathbf{j} - 21\mathbf{k}\]\[= (-21, -20, -21)\].
07

Solution Conclusion

After completing the cross product calculation, we find that the vector \( \mathbf{c} \times (\mathbf{a} + 3\mathbf{b}) = (-21, -20, -21) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cross Product
The cross product is an important operation in vector calculus, commonly used in physics and engineering. It is only defined for three-dimensional vectors and the result is a vector that is perpendicular to the plane formed by the original vectors.

Here is a simple breakdown of how a cross product works:
  • The cross product of two vectors \(\mathbf{u}\) and \(\mathbf{v}\) is denoted by \(\mathbf{u} \times \mathbf{v}\).
  • The magnitude of the resulting vector is equal to the area of the parallelogram formed by \(\mathbf{u}\) and \(\mathbf{v}\).
  • The direction of the cross product is determined using the right-hand rule: if your right fingers point from \(\mathbf{u}\) to \(\mathbf{v}\), your thumb points in the direction of \(\mathbf{u} \times \mathbf{v}\).

The algebraic calculation involves the determinant of a 3x3 matrix made from unit vectors and the components of the involved vectors. For instance, the cross product of our vectors \(\mathbf{c}\) and \(\mathbf{a} + 3\mathbf{b}\) is computed as the determinant of a matrix composed of these vectors.
Vector Addition
Vector addition is a fundamental operation where two or more vectors are combined to form a new vector, typically described as the sum of the individual vectors. It helps in simplifying problems in physics and engineering, such as forces and velocities.

The addition of vectors can be visualized geometrically:
  • Place the tail of the second vector at the head of the first vector.
  • The resultant vector (sum) is drawn from the tail of the first vector to the head of the second vector.
For our exercise, we first compute \(3\mathbf{b}\) by multiplying each component of \(\mathbf{b}\) by 3. We then simply add corresponding components of \(\mathbf{a}\) and \(3\mathbf{b}\). This gives us a new vector \((-6, -21, 26)\) which is the sum \(\mathbf{a} + 3\mathbf{b}\). It's important to ensure each component is accurately calculated to avoid errors.
Determinant Calculation
Determinant calculation is essential when dealing with matrices, and in this exercise, it provides a method for computing cross products. The determinant of a 3x3 matrix is calculated by expanding along a row, typically the top row, using a combination of the minors and cofactors of the matrix.

The process of finding the determinant involves:
  • Selecting a row or column to expand by (usually the first row for simplicity).

For example, to find the vector \(\mathbf{a}\) from its matrix, we compute the determinant:
  • Select the elements on the row: \(\mathbf{i}, \mathbf{j}, \mathbf{k}\)
  • Calculate each minor (the determinant of the sub-matrix that is not in the row or column of the element).
  • Multiply each minor by the related cofactor (\(\pm\) depending on its position).
This was employed to find \(\mathbf{a}\) and \(\mathbf{b}\), where we derived \((-9,0,5)\) and \((1,-7,7)\) respectively.
Step-by-step Solution
Breaking down problems into smaller steps is an effective way to tackle complicated mathematical operations, such as finding cross products. Each step helps guide you through the problem and clarifies the methodology.

  • Step 1 involves understanding the vectors \(\mathbf{a}, \mathbf{b},\) and \(\mathbf{c}\) by interpreting the problem.
  • Step 2 and 3 require finding \(\mathbf{a}\) and \(\mathbf{b}\) through determinant calculation.
  • In Step 4, we determine \(3\mathbf{b}\) before adding it to \(\mathbf{a}\) in Step 5.
  • Finally, Step 6 involves finding the cross product and concluding with the solution in Step 7.
This method not only simplifies complex operations but also serves as a guide for solving similar vector calculus problems. By organizing the solution in understandable steps, anyone can follow and replicate the process for other problems.

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