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Chapter 1: Problem 3

Cosine and sine by vector algebra* Find the cosine and the sine of the angle between \(\mathbf{A}=(3 \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}})\) and \(\mathbf{B}=(-2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}) .\)

Short Answer

Expert verified
\(\cos \theta = \frac{-4}{\sqrt{66}}\), \(\sin \theta = \sqrt{\frac{25}{33}}\).

Step by step solution

01

Calculate the Dot Product

The cosine of the angle between two vectors can be found using the dot product formula. First, calculate the dot product of \(\mathbf{A}\) and \(\mathbf{B}\): \(\mathbf{A} \cdot \mathbf{B} = (3)(-2) + (1)(1) + (1)(1) = -6 + 1 + 1 = -4\).
02

Calculate the Magnitude of the Vectors

Next, find the magnitude (norm) of both vectors \(\mathbf{A}\) and \(\mathbf{B}\). For \(\mathbf{A}\), \(\|\mathbf{A}\| = \sqrt{3^2 + 1^2 + 1^2} = \sqrt{11}\), and for \(\mathbf{B}\), \(\|\mathbf{B}\| = \sqrt{(-2)^2 + 1^2 + 1^2} = \sqrt{6}\).
03

Calculate Cosine of the Angle

The cosine of the angle \(\theta\) between \(\mathbf{A}\) and \(\mathbf{B}\) is \( \cos \theta = \frac{\mathbf{A} \cdot \mathbf{B}}{\|\mathbf{A}\| \|\mathbf{B}\|} = \frac{-4}{\sqrt{11} \cdot \sqrt{6}} = \frac{-4}{\sqrt{66}} \).
04

Calculate Sine of the Angle

To find the sine of the angle, we use the identity \( \sin^2 \theta + \cos^2 \theta = 1 \). Therefore, \( \sin \theta = \sqrt{1 - \cos^2 \theta} \). First, find \(\cos^2 \theta = \left( \frac{-4}{\sqrt{66}} \right)^2 = \frac{16}{66} = \frac{8}{33}\). So, \( \sin \theta = \sqrt{1 - \frac{8}{33}} = \sqrt{\frac{25}{33}}\).

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Key Concepts

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

Dot Product
In vector algebra, the dot product is a fundamental operation used to determine the angle between two vectors. It combines two vectors to produce a scalar, which gives an insight into how aligned the vectors are.

To find the dot product of two vectors, \( \mathbf{A} = a_1 \hat{\mathbf{i}} + a_2 \hat{\mathbf{j}} + a_3 \hat{\mathbf{k}} \) and \( \mathbf{B} = b_1 \hat{\mathbf{i}} + b_2 \hat{\mathbf{j}} + b_3 \hat{\mathbf{k}} \), you use the formula:
  • \( \mathbf{A} \cdot \mathbf{B} = a_1 b_1 + a_2 b_2 + a_3 b_3 \)
This formula calculates the sum of the products of the corresponding components of the vectors.

The dot product is especially useful in calculating the cosine of the angle between two vectors. Once you determine the dot product, the formula for cosine \( \theta \) is applied:
  • \( \cos \theta = \frac{\mathbf{A} \cdot \mathbf{B}}{\|\mathbf{A}\| \|\mathbf{B}\|} \)
This relation shows that the dot product is a crucial step in finding how two vectors interact based on their direction.
Magnitude of a Vector
The magnitude of a vector, often referred to as the vector's length or norm, is a measure of how long the vector is. It is always a non-negative scalar value.

To find the magnitude of a vector \( \mathbf{A} = a_1 \hat{\mathbf{i}} + a_2 \hat{\mathbf{j}} + a_3 \hat{\mathbf{k}} \), the formula is:
  • \( \|\mathbf{A}\| = \sqrt{a_1^2 + a_2^2 + a_3^2} \)
This results from applying the Pythagorean theorem in three dimensions. Calculating the magnitudes of vectors \( \mathbf{A} \) and \( \mathbf{B} \) is necessary for the formula to determine the cosine of the angle between them.

Understanding the magnitude of a vector is essential not only in geometry but also in physics and engineering, where it may represent quantities like displacement or force.
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that hold true for any value of the function's variables. These identities are key tools in expanding and simplifying expressions in algebra and calculus.

In this context, one primary identity used is the Pythagorean identity:
  • \( \sin^2 \theta + \cos^2 \theta = 1 \)
This fundamental identity helps in calculating the sine of the angle once the cosine is known. By rearranging the identity, we can find:
  • \( \sin \theta = \sqrt{1 - \cos^2 \theta} \)
This relation allows us to derive the sine from a known cosine, providing a complete understanding of the angle's trigonometric properties.

Mastery of these identities simplifies complex problems, making them a powerful tool in solving equations and interpreting the relationships between angles and sides in triangles.

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Most popular questions from this chapter

Perpendicular unit vectors \(^{*}\) Given vector \(\mathbf{A}=3 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}-4 \hat{\mathbf{k}}\), (a) find a unit vector \(\hat{B}\) that lies in the \(x-y\) plane and is perpendicular to A. (b) find a unit vector \(\hat{\mathbf{C}}\) that is perpendicular to both \(\mathbf{A}\) and \(\hat{\mathbf{B}}\). (c) Show that \(\mathbf{A}\) is perpendicular to the plane defined by \(\hat{\mathbf{B}}\) and \(\hat{\mathbf{C}}\)

Vector algebra \(1 \%\) Given two vectors \(\mathbf{A}=(2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+7 \hat{\mathbf{k}})\) and \(\mathbf{B}=(5 \hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}})\) find: (a) \(\mathbf{A}+\mathbf{B}\) (b) \(\mathbf{A}-\mathbf{B}\) (c) \(\mathbf{A} \cdot \mathbf{B}\) (d) \(\mathbf{A} \times \mathbf{B}\).

Volume of a parallelepiped Show that the volume of a parallelepiped with edges \(\mathbf{A}, \mathbf{B}\), and \(\mathbf{C}\) is given by \(\mathbf{A} \cdot(\mathbf{B} \times \mathbf{C})\).

Great circle \(^{*}\) The shortest distance between two points on the Earth (considered to be a perfect sphere of radius \(R\) ) is the distance along a great circle \(-\) the arc of a circle formed where a plane passing through the two points and the center of the Earth intersects the Earth's surface. The position of a point on the Earth is specified by the point's longitude \(\phi\) and latitude \(\lambda\). Longitude is the angle between the meridian (a line from pole to pole) passing through the point and the "prime" meridian passing through Greenwich U.K. Longitude is taken to be positive to the east and negative to the west. Latitude is the angle from the Equator along the point's meridian, taken positive to the north. Let the vectors from the center of the Earth to the points be \(\mathbf{r}_{1}\) and \(\mathbf{r}_{2}\). The cosine of the angle \(\theta\) between them can be found from their dot product, so that the great circle distance between the points is \(R \theta\). Find an expression for \(\theta\) in terms of the coordinates of the two points. Use a coordinate system with the \(x\) axis in the equatorial plane and passing through the prime meridian; let the \(z\) axis be on the polar axis, positive toward the north pole, as shown in the sketch.

Direction cosines The direction cosines of a vector are the cosines of the angles it makes with the coordinate axes. The cosines of the angles between the vector and the \(x, y\), and \(z\) axes are usually called, in turn, \(\alpha, \beta\), and \(\gamma .\) Prove that \(\alpha^{2}+\beta^{2}+\gamma^{2}=1\), using cither geometry or vector algebra.
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