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Chapter 3: Problem 74

Vector \(\vec{a}\) lies in the \(y z\) plane \(63.0^{\circ}\) from the positive direction of the \(y\) axis, has a positive \(z\) component, and has magnitude 3.20 units. Vector \(\vec{b}\) lies in the \(x z\) plane \(48.0^{\circ}\) from the positive direction of the \(x\) axis, has a positive \(z\) component, and has magnitude 1.40 units. Find (a) \(\vec{a} \cdot \vec{b},\) (b) \(\vec{a} \times \vec{b},\) and \((\mathrm{c})\) the angle between \(\vec{a}\) and \(\vec{b}\)

Short Answer

Expert verified
(a) 2.96, (b) \( \langle 1.51, -2.68, 1.36 \rangle \), (c) \( 48.5^\circ \).

Step by step solution

01

Identify the Components of Vector a

Given that vector \( \vec{a} \) lies in the \( yz \) plane and makes an angle of \( 63.0^\circ \) with the positive \( y \)-axis, we can resolve it into its components. Since the magnitude of \( \vec{a} \) is 3.20 units, the components are:\[ \vec{a} = 0 \hat{i} + 3.20 \cos(63^\circ) \hat{j} + 3.20 \sin(63^\circ) \hat{k} \]Calculating each component gives:\[ \vec{a} = 0 \hat{i} + 1.45 \hat{j} + 2.85 \hat{k} \]
02

Identify the Components of Vector b

Similarly, vector \( \vec{b} \) lies in the \( xz \) plane and makes an angle of \( 48.0^\circ \) with the positive \( x \)-axis. Hence, its components can be determined as:\[ \vec{b} = 1.40 \cos(48^\circ) \hat{i} + 0 \hat{j} + 1.40 \sin(48^\circ) \hat{k} \]Calculating each component gives:\[ \vec{b} = 0.94 \hat{i} + 0 \hat{j} + 1.04 \hat{k} \]
03

Compute the Dot Product \( \vec{a} \cdot \vec{b} \)

The dot product \( \vec{a} \cdot \vec{b} \) is computed using the formula:\[ \vec{a} \cdot \vec{b} = (a_i \cdot b_i) + (a_j \cdot b_j) + (a_k \cdot b_k) \]Substitute the values from the vectors:\[ \vec{a} \cdot \vec{b} = (0 \times 0.94) + (1.45 \times 0) + (2.85 \times 1.04) \]This results in:\[ \vec{a} \cdot \vec{b} = 0 + 0 + 2.96 = 2.96 \]
04

Compute the Cross Product \( \vec{a} \times \vec{b} \)

The cross product \( \vec{a} \times \vec{b} \) is found using the determinant of a matrix formed by the unit vectors \( \hat{i}, \hat{j}, \hat{k} \) and the components of \( \vec{a} \) and \( \vec{b} \):\[\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ 0 & 1.45 & 2.85 \ 0.94 & 0 & 1.04 \end{vmatrix}\]Calculating, we get:\[\vec{a} \times \vec{b} = \hat{i}(1.45 \cdot 1.04) - \hat{j}(0 \cdot 1.04) + \hat{k}(0 \cdot 0) - (2.85 \cdot 0.94)\]Simplifying further:\[ \vec{a} \times \vec{b} = (1.51 \hat{i} - 2.68 \hat{j} + 1.36 \hat{k}) \]
05

Calculate the Angle Between \( \vec{a} \) and \( \vec{b} \)

The angle \( \theta \) between two vectors is given by:\[ \cos \theta = \frac{\vec{a} \cdot \vec{b}}{\|\vec{a}\| \cdot \|\vec{b}\|} \]First, calculate the magnitudes:\[ \|\vec{a}\| = 3.20, \quad \|\vec{b}\| = 1.40 \]Substitute to find:\[ \cos \theta = \frac{2.96}{3.20 \times 1.40} \approx 0.663 \]Solving for \( \theta \), we find:\[ \theta = \cos^{-1}(0.663) \approx 48.5^\circ \]

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Key Concepts

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

Dot Product
The dot product is a way to multiply two vectors and get a scalar result. It measures how much one vector extends in the direction of another. When you compute the dot product, you multiply corresponding components of the vectors and take the sum:
  • For vectors \( \vec{a} \) and \( \vec{b} \), the dot product \( \vec{a} \cdot \vec{b} \) is calculated using:\[ a_i \cdot b_i + a_j \cdot b_j + a_k \cdot b_k \]
  • In our example, vectors \( \vec{a} \) and \( \vec{b} \) have been resolved into components as:\[ \vec{a} = 0 \hat{i} + 1.45 \hat{j} + 2.85 \hat{k} \]\[ \vec{b} = 0.94 \hat{i} + 0 \hat{j} + 1.04 \hat{k} \]
  • Substitute into the dot product formula:\[ \vec{a} \cdot \vec{b} = 0\cdot0.94 + 1.45\cdot0 + 2.85\cdot1.04 = 0 + 0 + 2.96 = 2.96 \]
The result 2.96 tells us how much of \( \vec{a} \) lies in the direction of \( \vec{b} \). If the vectors are perpendicular, their dot product is zero.
Cross Product
The cross product takes two vectors and produces another vector perpendicular to both, often used in physics to find a direction. It involves a determinant calculation of a matrix with unit vectors. Here鈥檚 how it works for vectors \( \vec{a} \) and \( \vec{b} \):
  • The matrix is:\[\begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ 0 & 1.45 & 2.85 \ 0.94 & 0 & 1.04 \end{vmatrix}\]
  • Calculate the components:
    • \( \hat{i} \) component: \( 1.45 \times 1.04 \)
    • There is no \( \hat{j} \) contribution from \( a_i \cdot b_k \)
    • \( \hat{k} \) component: \( 0 \cdot 0.94 - 2.85 \cdot 0 \)
  • The final vector is:\( \vec{a} \times \vec{b} = 1.51 \hat{i} - 0 \hat{j} + 1.36 \hat{k} \)
The cross product defines an axis direction with magnitude indicating rotational strength, highlighting vector differences or torque in physical systems.
Angle Between Vectors
Finding the angle between two vectors helps understand their geometric relationship. For vectors \( \vec{a} \) and \( \vec{b} \), use the cosine of the angle between them:
  • Formula for the cosine of the angle \( \theta \):\[ \cos \theta = \frac{\vec{a} \cdot \vec{b}}{\|\vec{a}\| \cdot \|\vec{b}\|} \]
  • Given magnitudes:\( \| \vec{a} \| = 3.20 \), and \( \| \vec{b} \| = 1.40 \)
  • Plug values into cosine formula:\[ \cos \theta = \frac{2.96}{3.20 \times 1.40} \approx 0.663 \]
  • Solve for \( \theta \):\[ \theta = \cos^{-1}(0.663) \approx 48.5^\circ \]
The calculated angle \( 48.5^\circ \) shows how the vectors \( \vec{a} \) and \( \vec{b} \) orient relative to each other in space.
Vector Components
Understanding vector components is vital for simplifying vector calculations. A vector is often split into components corresponding to each axis:
  • Vectors resolve into axis components using trigonometric functions of angles formed with the axes.
  • For a vector in the plane, components are found using:\[ \vec{v} = v_x \hat{i} + v_y \hat{j} + v_z \hat{k} \]
  • In this example, consider \( \vec{a} \):
    • Lies in the \( yz \) plane.
    • Components: \( 1.45 \hat{j} \) and \( 2.85 \hat{k} \)
  • For \( \vec{b} \):
    • Lies in the \( xz \) plane.
    • Components: \( 0.94 \hat{i} \) and \( 1.04 \hat{k} \)
Breaking down vectors into components simplifies operations like addition, dot, and cross products as you deal directly with one-dimensional values.

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Most popular questions from this chapter

(a) In unit-vector notation, what is the sum \(\vec{a}+\vec{b}\) if \(\vec{a}=(4.0 \mathrm{~m}) \hat{\mathrm{i}}+(3.0 \mathrm{~m}) \hat{\mathrm{k}}\) and \(\vec{b}=(-13.0 \mathrm{~m}) \hat{\mathrm{i}}+(7.0 \mathrm{~m}) \hat{\mathrm{k}} ?\) What are the (b) magnitude and (c) direction of \(\vec{a}+\vec{b}\) ?

You are to make four straight-line moves over a flat desert floor, starting at the origin of an \(x y\) coordinate system and ending at the \(x y\) coordinates \((-140 \mathrm{~m}, 30 \mathrm{~m}) .\) The \(x\) component and \(y\) component of your moves are the following, respectively, in meters: \((20\) and 60\(),\) then \(\left(b_{x}\right.\) and \(\left.-70\right),\) then \(\left(-20\right.\) and \(\left.c_{y}\right),\) then \((-60\) and -70\() .\) What are (a) component \(b_{x}\) and (b) component \(c_{y} ?\) What are (c) the magnitude and (d) the angle (relative to the positive direction of the \(x\) axis) of the overall displacement?

In the sum \(\vec{A}+\vec{B}=\vec{C},\) vector \(\vec{A}\) has a magnitude of \(12.0 \mathrm{~m}\) and is angled \(40.0^{\circ}\) counterclockwise from the \(+x\) direction, and vector \(\vec{C}\) has a magnitude of \(15.0 \mathrm{~m}\) and is angled \(20.0^{\circ}\) counterclockwise from the \(-x\) direction. What are (a) the magnitude and (b) the angle (relative to \(+x\) ) of \(\vec{B}\) ?

A protester carries his sign of protest, starting from the origin of an \(x y z\) coordinate system, with the \(x y\) plane horizontal. He moves \(40 \mathrm{~m}\) in the negative direction of the \(x\) axis, then \(20 \mathrm{~m}\) along a perpendicular path to his left, and then \(25 \mathrm{~m}\) up a water tower. (a) In unit-vector notation, what is the displacement of the sign from start to end? (b) The sign then falls to the foot of the tower. What is the magnitude of the displacement of the sign from start to this new end?

(a) What is the sum of the following four vectors in unitvector notation? For that sum, what are (b) the magnitude, (c) the angle in degrees, and (d) the angle in radians? \(\vec{E}: 6.00 \mathrm{~m}\) at \(+0.900 \mathrm{rad} \quad \vec{F}: 5.00 \mathrm{~m}\) at \(-75.0^{\circ}\) \(\vec{G}: 4.00 \mathrm{~m} \mathrm{at}+1.20 \mathrm{rad}\) \(\vec{H}: 6.00 \mathrm{~m}\) at \(-21.0^{\circ}\)
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