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Chapter 14: Problem 30

Prove the given property if \(\mathrm{a}=\left\langle a_{1}, a_{2}, a_{3}\right\rangle\) \(\mathrm{b}=\left\langle b_{1}, b_{2}, b_{3}\right\rangle, \mathrm{c}=\left\langle c_{1}, c_{2}, c_{3}\right\rangle,\) and \(m\) is a scalar. \(\quad(\mathbf{a} \times \mathbf{b}) \cdot \mathbf{c}=\mathbf{a} \cdot(\mathbf{b} \times \mathbf{c})\)

Short Answer

Expert verified
The property \((\mathbf{a} \times \mathbf{b}) \cdot \mathbf{c} = \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})\) is true.

Step by step solution

01

Express Cross Product

We need to express the cross products \( \mathbf{a} \times \mathbf{b} \) and \( \mathbf{b} \times \mathbf{c} \) in terms of components. The cross product \( \mathbf{a} \times \mathbf{b} \) is given by:\[ \mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \end{vmatrix} \]Expanding the determinant, we get:\[ \mathbf{a} \times \mathbf{b} = ((a_2b_3-a_3b_2), (a_3b_1-a_1b_3), (a_1b_2-a_2b_1)) \]
02

Express Second Cross Product

Similarly, express the second cross product \( \mathbf{b} \times \mathbf{c} \):\[ \mathbf{b} \times \mathbf{c} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ b_1 & b_2 & b_3 \ c_1 & c_2 & c_3 \end{vmatrix} \]Expanding this determinant, we get:\[ \mathbf{b} \times \mathbf{c} = ((b_2c_3-b_3c_2), (b_3c_1-b_1c_3), (b_1c_2-b_2c_1)) \]
03

Compute Dot Product of Cross Products

Now, compute the dot product of \( \mathbf{a} \times \mathbf{b} \) and \( \mathbf{c} \):\[(\mathbf{a} \times \mathbf{b}) \cdot \mathbf{c} = (a_2b_3 - a_3b_2)c_1 + (a_3b_1 - a_1b_3)c_2 + (a_1b_2 - a_2b_1)c_3 \]
04

Compute Dot Product for Second Expression

Similarly, compute the dot product of \( \mathbf{a} \) with \( \mathbf{b} \times \mathbf{c} \):\[\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = a_1(b_2c_3 - b_3c_2) + a_2(b_3c_1 - b_1c_3) + a_3(b_1c_2 - b_2c_1) \]
05

Verify Equality

Compare the expressions derived in Steps 3 and 4. Both are identical:- Expression from Step 3: \[(a_2b_3 - a_3b_2)c_1 + (a_3b_1 - a_1b_3)c_2 + (a_1b_2 - a_2b_1)c_3 \]- Expression from Step 4: \[a_1(b_2c_3 - b_3c_2) + a_2(b_3c_1 - b_1c_3) + a_3(b_1c_2 - b_2c_1) \]Therefore, \[ (\mathbf{a} \times \mathbf{b}) \cdot \mathbf{c} = \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \]. This proves the equality using the properties of vector triple products.

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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 a fundamental operation in vector calculus often used to find a vector that is perpendicular to two given vectors in three-dimensional space. Consider two vectors, \( \mathbf{a} = \langle a_1, a_2, a_3 \rangle \) and \( \mathbf{b} = \langle b_1, b_2, b_3 \rangle \). The cross product, denoted \( \mathbf{a} \times \mathbf{b} \), results in a new vector computed as follows:
  • The i-component is found by subtracting the product of the second component of \( \mathbf{a} \) and the third component of \( \mathbf{b} \) from the product of the third component of \( \mathbf{a} \) and the second component of \( \mathbf{b} \).
  • The j-component is calculated by subtracting the product of the third component of \( \mathbf{a} \) and the first component of \( \mathbf{b} \) from the product of the first component of \( \mathbf{a} \) and the third component of \( \mathbf{b} \).
  • The k-component comes from subtracting the product of the first component of \( \mathbf{a} \) and the second component of \( \mathbf{b} \) from the product of the second component of \( \mathbf{a} \) and the first component of \( \mathbf{b} \).
In essence, the cross product helps to determine volumes and orientations in space, and it's critical in physics for calculations involving torque and rotational forces.
Dot Product
The dot product, another essential operation in vector calculus, measures the extent to which two vectors point in the same direction. Unlike the cross product, the dot product results in a scalar, not a vector. To compute the dot product of two vectors, say \( \mathbf{u} = \langle u_1, u_2, u_3 \rangle \) and \( \mathbf{v} = \langle v_1, v_2, v_3 \rangle \), use the formula:
  • Multiply the corresponding components of the two vectors, giving \( u_1v_1, u_2v_2, \) and \( u_3v_3 \).
  • Sum these products to get the scalar \( \mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + u_3v_3 \).
The dot product is used in determining angles between vectors, calculating work done by a force, and in projections. If the dot product is zero, the vectors are orthogonal, meaning they are at a right angle to each other.
Triple Product Identity
The triple product identity involves both the dot product and the cross product, capturing the essence of three interacting vectors. Specifically, the identity \( (\mathbf{a} \times \mathbf{b}) \cdot \mathbf{c} = \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \) demonstrates an important symmetry in vector calculus. To understand it:
  • First, compute \( \mathbf{a} \times \mathbf{b} \), yielding a vector that is perpendicular to both \( \mathbf{a} \) and \( \mathbf{b} \).
  • Then take the dot product with \( \mathbf{c} \), effectively projecting \( \mathbf{c} \) onto this perpendicular vector.
  • Similarly, \( \mathbf{b} \times \mathbf{c} \) results in a vector perpendicular to \( \mathbf{b} \) and \( \mathbf{c} \), with the dot product \( \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \) projecting \( \mathbf{a} \) onto this vector.
This identity, known as the scalar triple product, is widely used in determining parallelism of vectors, calculating the volume of a parallelepiped, and in many engineering applications. It elegantly demonstrates how the interplay of vectors and their orientations follows consistent mathematical rules.

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Most popular questions from this chapter

Prove the given property if \(\mathrm{a}=\left\langle a_{1}, a_{2}\right\rangle,\) \(\mathrm{b}=\left\langle b_{1}, b_{2}\right\rangle, \mathrm{c}=\left\langle c_{1}, c_{2}\right\rangle,\) and \(p\) and \(q\) are real numbers. If \(p \mathbf{a}=0\) and \(\mathbf{a} \neq 0,\) then \(p=0\).

Sketch the graph of the equation in an \(x y z-\) coordinate system, and identify the surface. $$36 x^{2}-16 y^{2}+9 z^{2}=0$$

Sketch the graph of the quadric surface. Hyperbolic paraboloids \(|a| z=\frac{y^{2}}{9}-\frac{x^{2}}{4}\) (b) \(z=\frac{x^{2}}{4}-\frac{y^{2}}{9}\)

Describe the region \(R\) in a three-dimensional coordinate system. $$ R=\left\\{(x, y, z): x^{2}+y^{2} \leq 25,|z| \leq 3\right\\} $$

Solve without using components for the vectors. In the Cauchy-Schwarz inequality, under what conditions is \(|\mathbf{a} \cdot \mathbf{b}|=\|\mathbf{a}\|\|\mathbf{b}\| ?\)
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