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Chapter 7: Problem 23

(II) If $$\vec { \mathbf { A } } = 9.0 \hat { \mathbf { i } } - 8.5 \hat { \mathbf { j } } , \quad \vec { \mathbf { B } } = - 8.0 \hat { \mathbf { i } } + 7.1 \hat { \mathbf { j } } + 4.2 \hat { \mathbf { k } } , \quad$$ and $$\vec { \mathbf { C } } = 6.8 \hat { \mathbf { i } } - 9.2 \hat { \mathbf { j } } ,$$ determine $$( a ) \vec { \mathbf { A } } \cdot ( \vec { \mathbf { B } } + \vec { \mathbf { C } } ) ; ( b ) ( \vec { \mathbf { A } } + \vec { \mathbf { C } } ) \cdot \vec { \mathbf { B } }$$ $$( c ) ( \vec { \mathbf { B } } + \vec { \mathbf { A } } ) \cdot \vec { \mathbf { C } }$$

Short Answer

Expert verified
(a) 7.05, (b) -252.07, (c) 19.68.

Step by step solution

01

Calculate \( \vec{\mathbf{B}} + \vec{\mathbf{C}} \)

To determine \( \vec{\mathbf{B}} + \vec{\mathbf{C}} \), add the corresponding components of \( \vec{\mathbf{B}} \) and \( \vec{\mathbf{C}} \):\[ \vec{\mathbf{B}} + \vec{\mathbf{C}} = (-8.0 \hat{\mathbf{i}} + 6.8 \hat{\mathbf{i}}) + (7.1 \hat{\mathbf{j}} - 9.2 \hat{\mathbf{j}}) + (4.2 \hat{\mathbf{k}} + 0 \hat{\mathbf{k}}) \]Calculate each component:\[ = (-1.2 \hat{\mathbf{i}}) - (2.1 \hat{\mathbf{j}}) + 4.2 \hat{\mathbf{k}} \]
02

Compute \( \vec{\mathbf{A}} \cdot (\vec{\mathbf{B}} + \vec{\mathbf{C}}) \)

To compute the dot product \( \vec{\mathbf{A}} \cdot (\vec{\mathbf{B}} + \vec{\mathbf{C}}) \), use the formula \( \vec{\mathbf{A}} \cdot \vec{\mathbf{D}} = A_x D_x + A_y D_y + A_z D_z \):\[ \vec{\mathbf{A}} \cdot (\vec{\mathbf{B}} + \vec{\mathbf{C}}) = (9.0)(-1.2) + (-8.5)(-2.1) + (0)(4.2) \]Compute:\[ = -10.8 + 17.85 + 0 = 7.05 \]
03

Calculate \( \vec{\mathbf{A}} + \vec{\mathbf{C}} \)

To find \( \vec{\mathbf{A}} + \vec{\mathbf{C}} \), add the components of \( \vec{\mathbf{A}} \) and \( \vec{\mathbf{C}} \):\[ \vec{\mathbf{A}} + \vec{\mathbf{C}} = (9.0 + 6.8)\hat{\mathbf{i}} + (-8.5 - 9.2)\hat{\mathbf{j}} \]Calculate each component:\[ = 15.8 \hat{\mathbf{i}} - 17.7 \hat{\mathbf{j}} \]
04

Compute \( (\vec{\mathbf{A}} + \vec{\mathbf{C}}) \cdot \vec{\mathbf{B}} \)

Using the dot product formula, compute:\[ (\vec{\mathbf{A}} + \vec{\mathbf{C}}) \cdot \vec{\mathbf{B}} = (15.8)(-8.0) + (-17.7)(7.1) + (0)(4.2) \]Calculate:\[ = -126.4 - 125.67 + 0 = -252.07 \]
05

Calculate \( \vec{\mathbf{B}} + \vec{\mathbf{A}} \)

Find \( \vec{\mathbf{B}} + \vec{\mathbf{A}} \) by adding the components:\[ \vec{\mathbf{B}} + \vec{\mathbf{A}} = (-8.0 + 9.0)\hat{\mathbf{i}} + (7.1 - 8.5)\hat{\mathbf{j}} + 4.2 \hat{\mathbf{k}} \]Calculate each component:\[ = 1.0 \hat{\mathbf{i}} - 1.4 \hat{\mathbf{j}} + 4.2 \hat{\mathbf{k}} \]
06

Compute \( (\vec{\mathbf{B}} + \vec{\mathbf{A}}) \cdot \vec{\mathbf{C}} \)

Using the components calculated in the previous step, compute:\[ (\vec{\mathbf{B}} + \vec{\mathbf{A}}) \cdot \vec{\mathbf{C}} = (1.0)(6.8) + (-1.4)(-9.2) + (4.2)(0) \]Calculate:\[ = 6.8 + 12.88 + 0 = 19.68 \]

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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 in vector algebra. It's used to find the product of two vectors. The dot product has both mathematical and practical significance in measuring how much one vector goes in the direction of another:
  • Formula: For vectors \( \vec{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k} \) and \( \vec{B} = B_x \hat{i} + B_y \hat{j} + B_z \hat{k} \), their dot product is given by \( \vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z \).
  • Geometric Interpretation: The dot product also relates to the cosine of the angle between the vectors: \( \vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos(\theta) \), where \( \theta \) is the angle between \( \vec{A} \) and \( \vec{B} \).
  • Applications: It is used to project one vector onto another and calculate work done, given force and displacement vectors.
The dot product results in a scalar, not a vector. It shows how closely two vectors align by giving a positive, negative, or zero result based on their direction.
Vector Addition Explained
Vector addition is about combining vectors to find a resultant vector. This is akin to putting multiple directions together to find an overall path or displacement.
  • Component-wise Addition: To add two vectors \( \vec{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k} \) and \( \vec{B} = B_x \hat{i} + B_y \hat{j} + B_z \hat{k} \), you simply add their corresponding components: \( \vec{A} + \vec{B} = (A_x + B_x)\hat{i} + (A_y + B_y)\hat{j} + (A_z + B_z)\hat{k} \).
  • Visual Representation: You can visualize vector addition using the head-to-tail method. Place the tail of the second vector at the head of the first, and the resultant vector extends from the tail of the first to the head of the second.
  • Practical Use: Vector addition is crucial in physics for calculating net forces, velocity, and resultant displacements, especially when dealing with different direction vectors.
By understanding vector addition, students can effectively solve problems in mechanics and physics that involve multiple directions of forces or movements.
Understanding Vector Components
Vector components break a vector down into its fundamental parts along the coordinate axes, typically represented as \( x \), \( y \), and \( z \) axes for three-dimensional space.
  • Definition: Any vector \( \vec{V} \) can be expressed as \( \vec{V} = V_x \hat{i} + V_y \hat{j} + V_z \hat{k} \), where \( V_x \), \( V_y \), and \( V_z \) are its components along the respective axes.
  • Finding Components: The components can be found using trigonometry when given the magnitude \( |\vec{V}| \) and the angle it makes with each axis. For example, \( V_x = |\vec{V}| \cos(\alpha) \), where \( \alpha \) is the angle with the x-axis.
  • Importance: Vector components allow for simplified calculations of vector operations such as addition, subtraction, and the dot product. By treating each dimension independently, complex multi-dimensional problems become manageable.
Using vector components is a critical skill for breaking down and understanding problems in physics and engineering, where forces and movements often occur in multiple directions.

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Most popular questions from this chapter

Many cars have \({ }^{4} 5 \mathrm{mi} / \mathrm{h}(8 \mathrm{~km} / \mathrm{h})\) bumpers" that are designed to compress and rebound elastically without any physical damage at speeds below \(8 \mathrm{~km} / \mathrm{h}\). If the material of the bumpers permanently deforms after a compression of \(1.5 \mathrm{~cm}\), but remains like an elastic spring up to that point, what must be the effective spring constant of the bumper material, assuming the car has a mass of \(1050 \mathrm{~kg}\) and is tested by ramming into a solid wall?

As an object moves along the \(x\) axis from \(x = 0.0 \mathrm { m }\) to \(x = 20.0 \mathrm { m }\) it is acted upon by a force given by \(F = \left( 100 - ( x - 10 ) ^ { 2 } \right) \mathrm { N }\) . Determine the work done by the force on the object: \(( a )\) by first sketching the \(F\) vs. \(x\) graph and estimating the area under this curve; \(( b )\) by evaluating the integral \(\int _ { x = 0.0 \mathrm { m } } ^ { x = 20 \mathrm { m } } F d x\) .

What is the dot product of \(\overrightarrow{\mathbf{A}}=2.0 x^{2} \hat{\mathbf{i}}-4.0 x \hat{\mathbf{j}}+5.0 \hat{\mathbf{k}}\) and \(\overrightarrow{\mathbf{B}}=11.0 \hat{\mathbf{i}}+2.5 x \hat{\mathbf{j}}\) ?

A child is pulling a wagon down the sidewalk. For \(9.0 \mathrm{~m}\) the wagon stays on the sidewalk and the child pulls with a horizontal force of \(22 \mathrm{~N}\). Then one wheel of the wagon goes off on the grass so the child has to pull with a force of \(38 \mathrm{~N}\) at an angle of \(12^{\circ}\) to the side for the next \(5.0 \mathrm{~m}\). Finally the wagon gets back on the sidewalk so the child makes the rest of the trip, \(13.0 \mathrm{~m},\) with a force of \(22 \mathrm{~N}\). How much total work did the child do on the wagon?

Vector \(\mathbf{V}_{1}\) points along the \(z\) axis and has magnitude \(V_{1}=75 .\) Vector \(\overrightarrow{\mathbf{V}}_{2}\) lies in the \(x z\) plane, has magnitude \(V_{2}=58,\) and makes a \(-48^{\circ}\) angle with the \(x\) axis (points below \(x\) axis). What is the scalar product \(\overrightarrow{\mathbf{V}}_{1} \cdot \overrightarrow{\mathbf{v}}_{2} ?\)
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