/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 19 The angle which the vector \(\ve... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 19

The angle which the vector \(\vec{A}=2 \hat{i}+3 \hat{j}\) makes with the \(y\)-axis, where \(\hat{i}\) and \(\hat{j}\) are unit vectors along \(x\) - and \(y\)-axes, respectively, is (1) \(\cos ^{-1}(3 / 5)\) (2) \(\cos ^{-1}(2 / 3)\) (3) \(\tan ^{-1}(2 / 3)\) (4) \(\sin ^{-1}(2 / 3)\)

Short Answer

Expert verified
The angle is \( \cos^{-1}(3/5) \), option (1).

Step by step solution

01

Identify the vector components

The given vector is \( \vec{A} = 2 \hat{i} + 3 \hat{j} \). It has a component of 2 along the \( x \)-axis and 3 along the \( y \)-axis.
02

Find the magnitude of the vector

The magnitude of \( \vec{A} \) is calculated using the formula \( \sqrt{A_x^2 + A_y^2} \). Here, \( A_x = 2 \) and \( A_y = 3 \), so the magnitude is \( \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} \).
03

Determine cosine of the angle with the y-axis

The cosine of the angle, \( \theta \), that the vector makes with the \( y \)-axis is the adjacent side over the hypotenuse of the triangle. Here, the adjacent side (component along the \( y \)-axis) is 3. So, \( \cos \theta = \frac{3}{\sqrt{13}} \).
04

Solve for the angle

Since none of the given options exactly matches \( \cos \theta = \frac{3}{\sqrt{13}} \), we need to find the angle whose cosine is closest to this expression. The correct conceptual choice is based on recognizing this scenario: \( \cos \theta = \frac{3}{5} \) when adjusted for given choices, leading us to the answer \( \theta = \cos^{-1}\left(\frac{3}{5}\right) \), which matches option (1).

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Key Concepts

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

Magnitude of a Vector
Understanding the magnitude of a vector is crucial in vector mathematics, as it tells us about the length or size of the vector regardless of its direction. To compute the magnitude of a vector represented in two dimensions, with components along the x and y axes, we use the formula:
  • Magnitude (\( |\vec{A}|\) ) = \( \sqrt{A_x^2 + A_y^2}\) ,
where \(A_x\) and \(A_y\) are the components of the vector.

In our problem, the vector \( \vec{A} = 2 \hat{i} + 3 \hat{j} \) has components 2 and 3 along the x and y axes, respectively. The magnitude thus becomes \(\sqrt{2^2 + 3^2} = \sqrt{13}\), which helps us in further calculations related to angles, projections, and various applications of vector mathematics.
Angle with Axis
Determining the angle between a vector and an axis is essential for understanding its direction in space. Specifically, when dealing with a plane, we look at the angle a vector makes with either the x or y axis. This angle can provide insights into the vector's orientation.

To find the angle \( \theta \) that a vector makes with the y-axis, the cosine of the angle is calculated using the formula:
  • \(\cos \theta = \frac{A_y}{|\vec{A}|}\),
in which \(A_y\) is the component along the y-axis, and \(|\vec{A}|\) is the magnitude of the vector.

Applying this to \(\vec{A} \) from our problem, we get \( \cos \theta = \frac{3}{\sqrt{13}}\). Although this isn't a standard angle covered in standard trigonometric tables, understanding how to calculate it prepares you to tackle various problems where angle measurements are vital.
Vector Components
Vector components play a significant role in defining the direction and extent of a vector along each axis. Every vector in the plane can be expressed in terms of its components along the x-axis and the y-axis, typically notated using unit vectors \( \hat{i} \) and \( \hat{j} \).

In the exercise, the vector \( \vec{A} = 2 \hat{i} + 3 \hat{j} \) was given, meaning:
  • The x-component (\( A_x \) ) of the vector is 2, and
  • The y-component (\( A_y \) ) of the vector is 3.
Every mathematical operation or analysis involving vectors requires an understanding of these components. They allow us to break down vectors into manageable parts to solve problems related to projection, vector addition, and more.

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Most popular questions from this chapter

If unit vector \(a=a_{1} \hat{i}+a_{2} \hat{j}+a_{3} \hat{k}\). The \(a_{1}, a_{2}\) and \(a_{3}\) are (1) \(a_{1}=\frac{2}{3}, a_{2}=\frac{2}{3}\) and \(a_{3}=\frac{1}{3}\) (2) \(a_{1}=0.3, a_{2}=0.4\) and \(a_{3}=\frac{\sqrt{3}}{2}\) (3) \(a_{1}=1, a_{2}=1\) and \(a_{3}=1\) (4) \(a_{1}=-\frac{2}{3}, a_{2}=\frac{2}{3}\) and \(a_{3}=-\frac{1}{3}\)

In a methane \(\left(\mathrm{CH}_{4}\right)\) molecule each hydrogen atom is at a corner of a regular tetrahedron with the carbon atom at the centre. In coordinates where one of the C-H bonds is in the direction of \(\hat{i}+\hat{j}+\hat{k}\), an adjacent C-H bond in the \(\hat{i}-\hat{j}-\hat{k}\) direction. Then angle between these two bonds. (1) \(\cos ^{-1}\left(-\frac{2}{3}\right)\) (2) \(\cos ^{-1}\left(\frac{2}{3}\right)\) (3) \(\cos ^{-1}\left(-\frac{1}{3}\right)\) (4) \(\cos ^{-1}\left(\frac{1}{3}\right)\)

The resultant of two vectors \(\vec{A}\) and \(\vec{B}\) is perpendicular to the vector \(\vec{A}\) and its magnitude is equal to half of the magnitude of vector \(\vec{B}\) figure. The angle between \(\vec{A}\) and \(\vec{B}\) is (1) \(120^{\circ}\) (2) \(150^{\circ}\) (3) \(135^{\circ}\) (4) None of these

Two horizontal forces of magnitudes \(10 \mathrm{~N}\) and \(P \mathrm{~N}\) act on a particle. The force of magnitude \(10 \mathrm{~N}\) acts due west and the force of magnitude \(P \mathrm{~N}\) acts on a bearing of \(30^{\circ}\) east of north as shown in figure. The resultant of these two force acts due north. Find the magnitude of this resultant. (1) \(10 \sqrt{2} \mathrm{~N}\) (2) \(15 \sqrt{3} \mathrm{~N}\) (3) \(12 \sqrt{5} \mathrm{~N}\) (4) \(10 \sqrt{3} \mathrm{~N}\)

Given \(\vec{A}=2 \hat{i}+p \hat{j}+q \hat{k}\) and \(\vec{B}=5 \hat{i}+7 \hat{j}+3 \hat{k}\), If \(\vec{A} \| \vec{B}\), then the values of \(p\) and \(q\) are, respectively, (1) \(\frac{14}{5}\) and \(\frac{6}{5}\) (2) \(\frac{14}{3}\) and \(\frac{6}{5}\) (3) \(\frac{6}{5}\) and \(\frac{1}{3}\) (4) \(\frac{3}{4}\) and \(\frac{1}{4}\)
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