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Chapter 14: Problem 22

Find \(\mathbf{u} \cdot(\mathbf{v} \times \mathbf{w})\). $$u=\left(\begin{array}{l}3 \\\2 \\\1\end{array}\right), \mathbf{v}=\left(\begin{array}{c}1 \\\\-3 \\\1\end{array}\right), \mathbf{w}=\left(\begin{array}{l}5 \\\1 \\\2\end{array}\right)$$

Short Answer

Expert verified
The value is \(-11\).

Step by step solution

01

Understand the Problem

We need to find the dot product of vector \( \mathbf{u} \) with the cross product of vectors \( \mathbf{v} \) and \( \mathbf{w} \). This is expressed as \( \mathbf{u} \cdot(\mathbf{v} \times \mathbf{w}) \).
02

Calculate the Cross Product \( \mathbf{v} \times \mathbf{w} \)

The cross product is given by:\[ \mathbf{v} \times \mathbf{w} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ 1 & -3 & 1 \ 5 & 1 & 2 \end{vmatrix} \]This results in:- \( \mathbf{i}( (-3)(2) - (1)(1) ) = \mathbf{i}(-6 - 1) = -7\mathbf{i} \)- \( \mathbf{j}( (2)(1) - (1)(5) ) = \mathbf{j}(2 - 5) = -3\mathbf{j} \)- \( \mathbf{k}( (1)(1) - (-3)(5) ) = \mathbf{k}(1 + 15) = 16\mathbf{k} \)Thus, \( \mathbf{v} \times \mathbf{w} = \left(-7, -3, 16\right) \).
03

Calculate the Dot Product \( \mathbf{u} \cdot(\mathbf{v} \times \mathbf{w}) \)

Now calculate the dot product using:\[ (3, 2, 1) \cdot (-7, -3, 16) = 3(-7) + 2(-3) + 1(16) \]Calculate each term:- \( 3(-7) = -21 \)- \( 2(-3) = -6 \)- \( 1(16) = 16 \)Add these up:\[ -21 - 6 + 16 = -11 \]
04

Conclusion

The dot product \( \mathbf{u} \cdot(\mathbf{v} \times \mathbf{w}) \) is \(-11\).

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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 mathematical operation performed on two vectors in three-dimensional space. It results in another vector that is orthogonal, or perpendicular, to both of the original vectors. Let's explore this in more detail to make this clear.

  • The formula to calculate the cross product of two vectors, say \( \mathbf{v} = (v_1, v_2, v_3) \) and \( \mathbf{w} = (w_1, w_2, w_3) \), involves finding a determinant:
\[\mathbf{v} \times \mathbf{w} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ v_1 & v_2 & v_3 \ w_1 & w_2 & w_3 \end{vmatrix}\]- The result is computed by determinant methods and leads to a vector with components:

  • \( ((v_2 w_3 - v_3 w_2)\mathbf{i}) - ((v_1 w_3 - v_3 w_1)\mathbf{j}) + ((v_1 w_2 - v_2 w_1)\mathbf{k}) \)
The cross product is particularly useful in physics and engineering contexts where perpendicularity is desired, such as calculating torques or angular momentum.
Dot Product
The dot product, often referred to as the scalar product, is an operation that produces a scalar (a single number) from two vectors. It is a vital concept in vector calculus because it allows measuring how much one vector "projects" onto another. Let's unpack this concept.

  • Sometimes, people think about dot products in terms of angles: the dot product helps determine if two vectors are parallel, orthogonal, or neither.
  • The formula for the dot product between vectors \( \mathbf{a} = (a_1, a_2, a_3) \) and \( \mathbf{b} = (b_1, b_2, b_3) \) is:
\[\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3\]

  • The result is a single number that signifies the magnitude of one vector in the direction of the other.
  • In practical situations, this helps in finding the angle between the vectors or in understanding concepts such as work done by a force when moving an object.
Mastering the dot product concept helps tremendously in fields involving directional data, like computer graphics and physical systems.
Vectors in Mathematics
Vectors are fundamental objects in mathematics and physics that possess both a magnitude and a direction. They are essential for various mathematical computations and physical scenarios.

  • Typically represented as ordered pairs or triples in two-dimensional or three-dimensional space, vectors are the building blocks of vector calculus.
  • One of the simplest representations of a vector in a 3D space is \( \mathbf{a} = (a_1, a_2, a_3) \).
In exploring vector operations like addition, subtraction, dot product, and cross product, vectors allow combinations of directions and magnitudes, which are key in physical applications like force or velocity.

  • Vectors are extensively used in disciplines ranging from physics and engineering to computer science, particularly in modeling real-world phenomena where direction and magnitude are critical.
  • Understanding vectors and their operations allows for solving complex problems intuitively and effectively.
Thus, vectors provide an invaluable framework for both theoretical research and practical applications, making them indispensable in mathematics and beyond.

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Most popular questions from this chapter

What point on the line through (1,2,5) and (3,1,1) is closest to the point (2,-1,5)\(?\)

Consider the points \(A(1,3,-17)\) and \(B(6,-7,8)\) which lie on the line \(1 .\) a) Find an equation of line \(l,\) giving the answer in parametric form. b) The point \(P\) is on \(/\) such that \(\overrightarrow{O P}\) is perpendicular to \(/\). Find the coordinates of \(P\).

Determine the values of \(m\) and \(n\) such that the vectors \(\mathbf{v}(m-2, m+n,-2 m+n)\) and \(\mathbf{w}(2,4,-6)\) have the same direction.

Find a unit vector that is normal (perpendicular) to the plane determined by the points \(A(1,-1,2), B(2,0,-1)\) and \(C(0,2,1)\).

Find the acute angle between the given lines or planes. $$4 x-7 y+z=3 \text { and } 3 x+2 y+2 z=17$$
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