Question

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

Short answer

Additionally, if u and v both make a 90° angle with w, does their sum (u+v) always make a 90° angle with w? Also, does the sum of the vectors (1,0,0) + (0,1,0) + (0,0,1) equal the zero vector? Lastly, does the intersection of the planes x=1, y=1, and z=1 consist of a single point? Answer: No, it is not always true that when vectors u and v both make a 45° angle with vector w, their sum (u+v) also makes a 45° angle with w. It is also not always true that when u and v both make a 90° angle with w, their sum (u+v) always makes a 90° angle with w. Additionally, the sum of the vectors (1,0,0) + (0,1,0) + (0,0,1) is not the zero vector ((0,0,0)), as their sum equals the vector (1,1,1). Lastly, the intersection of the planes x=1, y=1, and z=1 does consist of a single point, which is the point (1,1,1).

Step-by-step solution

a. Checking angles in \(\mathrm{R}^{3}\)#

Given that \(\mathbf{u}\) and \(\mathbf{v}\) both make a \(45^{\circ}\) angle with \(\mathbf{w}\), we need to check if their sum, \(\mathbf{u}+\mathbf{v}\), also makes a \(45^{\circ}\) angle with \(\mathbf{w}\). This statement is false. Here is a counterexample: let \(\mathbf{u}=(1,0,1)\), \(\mathbf{v}=(0,1,1)\), and \(\mathbf{w}=(1,1,0)\). We can check that both \(\angle(\mathbf{u},\mathbf{w})=45^{\circ}\) and \(\angle(\mathbf{v},\mathbf{w})=45^{\circ}\). However, \(\mathbf{u}+\mathbf{v}=(1,1,2)\) does not make a \(45^{\circ}\) angle with \(\mathbf{w}\).

b. Verifying orthogonal vectors in \(\mathrm{R}^{3}\)#

Given that \(\mathbf{u}\) and \(\mathbf{v}\) both make a \(90^{\circ}\) angle with \(\mathbf{w}\), which means they are orthogonal, we need to verify if their sum, \(\mathbf{u}+\mathbf{v}\), is also orthogonal to \(\mathbf{w}\) (makes a \(90^{\circ}\) angle). This statement is false. Here is a counterexample: let \(\mathbf{u}=(1,0,-1)\), \(\mathbf{v}=(-1,0,-1)\), and \(\mathbf{w}=(0,1,0)\). We can check that both \(\angle(\mathbf{u},\mathbf{w})=90^{\circ}\) and \(\angle(\mathbf{v},\mathbf{w})=90^{\circ}\). However, \(\mathbf{u}+\mathbf{v}=(0,0,-2)\) does make a \(90^{\circ}\) angle with \(\mathbf{w}\).

c. Checking vector sum#

We need to check if \(\overline{1}+j+k = 0\). In this case, \(\overline{1}\) is the vector scaled by \(1\) along the \(x\)-axis, \(j\) is the unit vector along the \(y\)-axis, and \(k\) is the unit vector along the \(z\)-axis. Their sum is: \(\overline{1} + j + k = (1,0,0) + (0,1,0) + (0,0,1) = (1,1,1)\) Since \((1,1,1) \neq (0,0,0)\), the statement is false.

d. Finding the intersection of planes#

We need to find the intersection of the planes \(x=1\), \(y=1\), and \(z=1\). Let's find their intersections one at a time: Intersection of \(x=1\) and \(y=1\): This gives us a line in the form \((1,1,z)\), where \(z\) is any real number. Intersection of the line \((1,1,z)\) and the plane \(z=1\): This point of intersection is \((1,1,1)\). The intersection of the three planes is a point, \((1,1,1)\). So, the statement is true.

Key concepts

Vectors in R3

When we talk about vectors in \( ext{R}^3\), we're diving into a topic that deals with vectors having three components, each corresponding to the x, y, and z axes in a three-dimensional space. Vectors are essentially points in this space with direction and magnitude. They can be represented as \(\mathbf{v} = (x, y, z)\). Each component represents how much of the vector is pointing in each of these three axis directions.
You can imagine vectors as arrows that start at the origin of this space and point to the coordinates \(x, y, z\).
In many problems, like those involving physical forces, vectors help us understand diagonal forces acting simultaneously in different directions.

Orthogonal Vectors

Two vectors are orthogonal if they form a 90-degree angle with each other. This concept is common when dealing with vector calculus in \(\text{R}^3\). Because of the angle, orthogonal vectors have a dot product equal to zero. The dot product of two vectors \(\mathbf{a} = (a_1, a_2, a_3)\) and \(\mathbf{b} = (b_1, b_2, b_3)\) is given by:
\( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \)
If \(\mathbf{a} \) and \(\mathbf{b} \) are orthogonal, this expression equals zero. This means that the two vectors don't point towards or away from each other at all; they are completely independent in direction.
Orthogonality is crucial in many calculations and applications, such as simplifying problems by considering independent vectors.

Intersection of Planes

In vector calculus, planes in a 3D space can intersect each other in various ways. The intersection could be a line or even just a single point, depending on the orientations and positions of the planes involved.
A plane in \(\text{R}^3\) is typically described with an equation that looks like \(ax + by + cz = d\), where \(a, b,\) and \(c\) are coefficients that determine the plane's orientation.
When examining the intersection of multiple planes, it's important to find where all these equations are satisfied simultaneously. An interesting scenario occurs when three planes intersect at just one point, as in the given problem: \(x=1\), \(y=1\), and \(z=1\). This means they've converged at a single common point \(\left(1, 1, 1\right)\).
Understanding plane intersections is fundamental in geometry and helpful in solving real-world navigation and construction problems.