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Chapter 11: Problem 57

Determine whether the following statements are true and give an explanation or counterexample. a. Suppose \(\mathbf{u}\) and \(\mathbf{v}\) both make a \(45^{\circ}\) angle with \(\mathbf{w}\) in \(\mathrm{R}^{3}\). Then \(\mathbf{u}+\mathbf{v}\) makes a \(45^{\circ}\) angle with \(\mathbf{w}\) b. Suppose \(\mathbf{u}\) and \(\mathbf{v}\) both make a \(90^{\circ}\) angle with \(\mathbf{w}\) in \(\mathbb{R}^{3}\). Then \(\mathbf{u}+\mathbf{v}\) can never make a \(90^{\circ}\) angle with \(\mathbf{w}\) c. \(i+j+k=0\) d. The intersection of the planes \(x=1, y=1,\) and \(z=1\) is a point.

Short Answer

Expert verified
Provide explanations or counterexamples for each statement. a) If vector \(\mathbf{u}\) and vector \(\mathbf{v}\) both make a \(45^{\circ}\) angle with vector \(\mathbf{w}\), then the vector \(\mathbf{u}+\mathbf{v}\) will also make a \(45^{\circ}\) angle with vector \(\mathbf{w}\). Answer: False. Explanation: Consider the vectors \(\mathbf{u}=(1,0,0)\), \(\mathbf{v}=(0,1,0)\), and \(\mathbf{w}=(1,1,0)\). Both \(\mathbf{u}\) and \(\mathbf{v}\) make a \(45^{\circ}\) angle with \(\mathbf{w}\). However, the angle between \(\mathbf{u}+\mathbf{v}\) and \(\mathbf{w}\) is not \(45^{\circ}\), but instead \(0^{\circ}\). b) If vector \(\mathbf{u}\) and vector \(\mathbf{v}\) both make a \(90^{\circ}\) angle with vector \(\mathbf{w}\), then the vector \(\mathbf{u}+\mathbf{v}\) can never make a \(90^{\circ}\) angle with vector \(\mathbf{w}\). Answer: False. Explanation: Consider the vectors \(\mathbf{u}=(1,0,0)\), \(\mathbf{v}=(0,1,0)\), and \(\mathbf{w}=(0,0,1)\). Both \(\mathbf{u}\) and \(\mathbf{v}\) make a \(90^{\circ}\) angle with \(\mathbf{w}\), but the angle between \(\mathbf{u}+\mathbf{v}\) and \(\mathbf{w}\) is also \(90^{\circ}\). c) The sum of the three unit vectors \(i\), \(j\), and \(k\) is equal to zero. Answer: False. Explanation: If we add \(i + j + k\), we get a vector \((1,1,1) \neq 0\). d) The geometric figure created by the intersection of the planes \(x=1\), \(y=1\), and \(z=1\) is a point. Answer: True. Explanation: The three planes intersect at the point \((x,y,z) = (1,1,1)\).

Step by step solution

01

Recalling the angle formula.

The angle between two vectors \(\mathbf{u}\) and \(\mathbf{v}\) can be calculated using the dot product: \(\cos \theta = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{u}||\mathbf{v}|}\)
02

Consider the given vectors.

Vector \(\mathbf{u}\) and vector \(\mathbf{v}\) both make a \(45^{\circ}\) angle with vector \(\mathbf{w}\). We have to check if the vector \(\mathbf{u}+\mathbf{v}\) makes a \(45^{\circ}\) angle with vector \(\mathbf{w}\).
03

Counterexample.

Let's consider the vectors \(\mathbf{u}=(1,0,0)\), \(\mathbf{v}=(0,1,0)\), and \(\mathbf{w}=(1,1,0)\). Both \(\mathbf{u}\) and \(\mathbf{v}\) make a \(45^{\circ}\) angle with \(\mathbf{w}\). However, if we add \(\mathbf{u}\) and \(\mathbf{v}\), we get a new vector \((1,1,0)\) which is parallel to \(\mathbf{w}\). In this case, the angle between \(\mathbf{u}+\mathbf{v}\) and \(\mathbf{w}\) is not \(45^{\circ}\), but instead \(0^{\circ}\). This counterexample contradicts the statement. For statement a, the answer is False. For statement b:
04

Recalling the angle formula.

The angle between two vectors \(\mathbf{u}\) and \(\mathbf{v}\) can be calculated using the dot product: \(\cos \theta = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{u}||\mathbf{v}|}\)
05

Consider the given vectors.

Vector \(\mathbf{u}\) and vector \(\mathbf{v}\) both make a \(90^{\circ}\) angle with vector \(\mathbf{w}\). We have to check if the vector \(\mathbf{u}+\mathbf{v}\) can never make a \(90^{\circ}\) angle with vector \(\mathbf{w}\).
06

Counterexample.

Let's consider the vectors \(\mathbf{u}=(1,0,0)\), \(\mathbf{v}=(0,1,0)\), and \(\mathbf{w}=(0,0,1)\). Both \(\mathbf{u}\) and \(\mathbf{v}\) make a \(90^{\circ}\) angle with \(\mathbf{w}\). However, if we add \(\mathbf{u}\) and \(\mathbf{v}\), we get a new vector \((1,1,0)\) which also makes a \(90^{\circ}\) angle with \(\mathbf{w}\). This contradicts the statement. For statement b, the answer is False. For statement c:
07

Understanding the symbols.

Here, \(i\), \(j\), and \(k\) represent the unit vectors in the \(x\), \(y\), and \(z\) directions respectively.
08

Adding the unit vectors.

If we add \(i + j + k\), we get a vector \((1,1,1) \neq 0\). Thus, the statement is false. For statement c, the answer is False. For statement d:
09

Identify planes.

The given planes are defined as follows: \(x=1\), \(y=1\), and \(z=1\).
10

Calculate intersection.

In order to find the intersection of three planes, we solve the equations for each plane simultaneously. Since the three planes are in the form of constants, it's easier to see that the intersection point is at \((x,y,z) = (1,1,1)\). For statement d, the answer is True.

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Key Concepts

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

Dot Product
Understanding the dot product is essential in calculating the angle between two vectors. The dot product, also known as the scalar product, is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. This operation is defined by
\[\begin{equation}\mathbf{u} \cdot \mathbf{v} = |\mathbf{u}||\mathbf{v}|\cos(\theta)\end{equation}\]
where \(\mathbf{u}\) and \(\mathbf{v}\) are vectors, \(|\mathbf{u}|\) and \(|\mathbf{v}|\) are the magnitudes (lengths) of the vectors, and \(\theta\) is the angle between them. The dot product has geometric interpretations; for instance, when the dot product is zero, the vectors are perpendicular. In the context of angles, the dot product is used to find the cosine of the angle between two vectors, which can then be used to determine the angle itself. Thus, mastery of the dot product paves the way to a deeper understanding of vector angles and spatial relationships.
Vector Addition
Moving on to vector addition, this operation plays a crucial role in vector analysis and physics. The process involves adding corresponding components of two or more vectors to create a new vector. If considering vectors as arrows with direction and magnitude, the addition is like placing the tail of one arrow to the head of the other and drawing a vector from the free tail to the free head鈥攖hat's your resulting vector. Mathematically, it's expressed as:
\[\begin{equation}\mathbf{u} + \mathbf{v} = (u_1 + v_1, u_2 + v_2, u_3 + v_3)\end{equation}\]
Vector addition is associative and commutative, meaning the order of addition does not change the result. In the context of our original problem, although two vectors might individually form the same angle with a third vector, their sum does not necessarily maintain that angle due to the properties of vector addition. This is illustrated by the provided counterexample.
Counterexample in Mathematics
A counterexample is a powerful concept used to disprove statements or conjectures in mathematics. It is an instance that shows a general statement is not always true. To invalidate a proposition that claims 'all X are Y', one only needs to find a single case where 'X is not Y.' Counterexamples play a fundamental role in the process of mathematical proof and discovery.
In our textbook problem, counterexamples were constructed to demonstrate the falseness of particular propositions about vector angles. These counterexamples were effective because they provided specific, clear instances where the original statements did not hold. Identifying appropriate counterexamples requires not only creativity but a robust understanding of the mathematical concepts involved.
Unit Vectors
Lastly, let's delve into unit vectors. A unit vector is a vector that has a magnitude of one. In a coordinate system, unit vectors are used to represent the direction of axes. For a three-dimensional Cartesian coordinate system, the standard unit vectors are \(i\), \(j\), and \(k\), which point in the direction of the positive x, y, and z-axis, respectively. They are expressed as:
\[\begin{equation}\mathbf{i} = (1,0,0), \mathbf{j} = (0,1,0), \mathbf{k} = (0,0,1)\end{equation}\]
Unit vectors are essential in physics and engineering as they are the building blocks used to describe directions and orientations in space. They also help clarify statements and equations by emphasizing direction without magnitude. The problem statement involving \(i + j + k\) is a vector addition of unit vectors, which erroneously suggests that their sum could be a zero vector, but instead, it creates another vector of equal magnitude in all directions from the origin.

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Most popular questions from this chapter

Use vectors to show that the midpoint of the line segment joining \(P\left(x_{1}, y_{1}\right)\) and \(Q\left(x_{2}, y_{2}\right)\) is the point \(\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right)\) (Hint: Let \(O\) be the origin and let \(M\) be the midpoint of \(P Q\). Draw a picture and show that $$\left.\overrightarrow{O M}=\overrightarrow{O P}+\frac{1}{2} \overrightarrow{P Q}=\overrightarrow{O P}+\frac{1}{2}(\overrightarrow{O Q}-\overrightarrow{O P}) \cdot\right)$$

Compute the following derivatives. $$\frac{d}{d t}\left(\left(t^{3} \mathbf{i}-2 t \mathbf{j}-2 \mathbf{k}\right) \times\left(t \mathbf{i}-t^{2} \mathbf{j}-t^{3} \mathbf{k}\right)\right)$$

Show that the (least) distance \(d\) between a point \(Q\) and a line \(\mathbf{r}=\mathbf{r}_{0}+t \mathbf{v}\) (both in \(\mathbb{R}^{3}\) ) is \(d=\frac{|\overrightarrow{P Q} \times \mathbf{v}|}{|\mathbf{v}|},\) where \(P\) is a point on the line.

Compute \(\mathbf{r}^{\prime \prime}(t)\) and \(\mathbf{r}^{\prime \prime \prime}(t)\) for the following functions. $$\mathbf{r}(t)=\left\langle 3 t^{12}-t^{2}, t^{8}+t^{3}, t^{-4}-2\right\rangle$$

Use the formula in Exercise 79 to find the (least) distance between the given point \(Q\) and line \(\mathbf{r}\). $$Q(5,6,1) ; \mathbf{r}(t)=\langle 1+3 t, 3-4 t, t+1\rangle$$
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