Problem 63
Question
A line \(\mathrm{AB}\) in 3 -dimensional space makes angles \(45^{\circ}\) and \(120^{\circ}\) with the positive \(x\)-axis and the positive \(y\)-axis respectively. If \(A B\) makes an acute angle \(\theta\) with the positive \(z\)-axis, then \(\theta\) equals [2010] (A) \(45^{\circ}\) (B) \(60^{\circ}\) (C) \(75^{\circ}\) (D) \(30^{\circ}\)
Step-by-Step Solution
Verified Answer
\(\theta = 60^{\circ}\) (Option B)
1Step 1: Recall the Direction Cosines Concept
In 3-dimensional geometry, a line can be characterized by its angles with the positive x, y, and z axes. These angles are denoted by \(\alpha\), \(\beta\), and \(\theta\) respectively. The cosine of these angles give us direction cosines \(l\), \(m\), and \(n\), where \(l = \cos \alpha\), \(m = \cos \beta\), and \(n = \cos \theta\).
2Step 2: Apply the Direction Cosine Equation
For a line in 3D space, the sum of the squares of its direction cosines equals 1. Mathematically, it is represented as: \[ l^2 + m^2 + n^2 = 1 \].
3Step 3: Substitute Known Values
The problem states \(\alpha = 45^{\circ}\) and \(\beta = 120^{\circ}\). We find the direction cosines:\(l = \cos 45^{\circ} = \frac{1}{\sqrt{2}}\) and \(m = \cos 120^{\circ} = -\frac{1}{2}\).
4Step 4: Solve for \(n\)
Substitute the known values in the direction cosine equation:\[\left(\frac{1}{\sqrt{2}}\right)^2 + \left(-\frac{1}{2}\right)^2 + n^2 = 1\]Calculate each term:\[\left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2}, \quad \left(-\frac{1}{2}\right)^2 = \frac{1}{4}\]Thus, \[\frac{1}{2} + \frac{1}{4} + n^2 = 1\] Simplifying, \[\frac{3}{4} + n^2 = 1\] Thus,\[n^2 = \frac{1}{4}\] Take the positive square root since \(\theta\) is acute:\[n = \frac{1}{2}\]
5Step 5: Find \(\theta\) Using the Inverse Cosine
Since \(n = \cos \theta\) and \(n = \frac{1}{2}\), use the inverse cosine to solve for \(\theta\):\[\theta = \cos^{-1}\left(\frac{1}{2}\right)\]This results in, \(\theta = 60^{\circ}\).
Key Concepts
3-dimensional geometrydirection cosine equationangles with axes
3-dimensional geometry
In 3-dimensional geometry, understanding how lines function in a 3D space is crucial. We explore how a line in space can align itself with the three main axes: the x-axis, y-axis, and z-axis. The line, typically represented as AB in exercises, will make certain angles with these axes. This spatial understanding is foundational in mathematics and applies broadly in fields like physics and engineering.
One might wonder why engineers and architects often rely on 3D geometry. Well, this realm of geometry helps in modeling real-world structures and objects since most things in our world have length, width, and height. When working with 3D models, knowing the positioning and alignment of elements against these axes becomes powerful. It's like finding the line of best fit, but in a full 3D space.
We'll use such concepts as direction cosines and angles with axes to determine how lines position themselves in a 3-dimensional realm.
One might wonder why engineers and architects often rely on 3D geometry. Well, this realm of geometry helps in modeling real-world structures and objects since most things in our world have length, width, and height. When working with 3D models, knowing the positioning and alignment of elements against these axes becomes powerful. It's like finding the line of best fit, but in a full 3D space.
We'll use such concepts as direction cosines and angles with axes to determine how lines position themselves in a 3-dimensional realm.
direction cosine equation
When dealing with 3-dimensional problems, direction cosines become particularly useful. These are the cosines of the angles a line makes with the x, y, and z axes. Denoted typically by \(l\), \(m\), and \(n\), these cosines help describe the line's orientation in space.
The key equation involving direction cosines is the direction cosine equation, which states:
Direction cosines hold an important relationship, described by the equation:\[ l^2 + m^2 + n^2 = 1 \]This equation expresses that the sum of the squares of the direction cosines equals 1. It's this relationship that enables us to solve various problems involving lines and angles in 3D geometry—balanced much like the Pythagorean theorem but extended into a more complex, dimensional context. Understanding this equation is key to solving many geometric problems, as it ties the line's direction components on all three axes together.
The key equation involving direction cosines is the direction cosine equation, which states:
- \(l = \cos \alpha\)
- \(m = \cos \beta\)
- \(n = \cos \theta\)
Direction cosines hold an important relationship, described by the equation:\[ l^2 + m^2 + n^2 = 1 \]This equation expresses that the sum of the squares of the direction cosines equals 1. It's this relationship that enables us to solve various problems involving lines and angles in 3D geometry—balanced much like the Pythagorean theorem but extended into a more complex, dimensional context. Understanding this equation is key to solving many geometric problems, as it ties the line's direction components on all three axes together.
angles with axes
Angles made with axes are pivotal in defining the orientation of a line in 3-dimensional space. When a line forms an angle with an axis, it agrees with that axis over a specific region in space, making the concept of angles essential for visualizing spatial relationships.
In our exercise, the line AB creates angles of \(45^{\circ}\) and \(120^{\circ}\) with the x and y axes, respectively. The challenge is to determine its angle with the z-axis, known as \(\theta\). To solve for \(\theta\), we use the calculated direction cosines, which, when squared and summed, equals 1 as per the direction cosine equation. This solution cleverly uses trigonometry to decode the harmonious spatial alignment of lines.
By extracting one angle using inverse functions, we gather a holistic understanding of how the line positions itself. Understanding these angles helps us not only in math but also when engaging with practical visualization tasks, like designing or interpreting objects in software that models 3D environments.
In our exercise, the line AB creates angles of \(45^{\circ}\) and \(120^{\circ}\) with the x and y axes, respectively. The challenge is to determine its angle with the z-axis, known as \(\theta\). To solve for \(\theta\), we use the calculated direction cosines, which, when squared and summed, equals 1 as per the direction cosine equation. This solution cleverly uses trigonometry to decode the harmonious spatial alignment of lines.
By extracting one angle using inverse functions, we gather a holistic understanding of how the line positions itself. Understanding these angles helps us not only in math but also when engaging with practical visualization tasks, like designing or interpreting objects in software that models 3D environments.
Other exercises in this chapter
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