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Chapter 4: Problem 32

If \(\vec{d}_{1}+\vec{d}_{2}=5 \vec{d}_{3}, \vec{d}_{1}-\vec{d}_{2}=3 \vec{d}_{3}\), and \(\vec{d}_{3}=2 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}\), then what are (a) \(\vec{d}_{1}\) and (b) \(\vec{d}_{2}\) ?

Short Answer

Expert verified
\(\vec{d}_{1} = 8\hat{i} + 16\hat{j}\) and \(\vec{d}_{2} = 2\hat{i} + 4\hat{j}\)

Step by step solution

01

Substitute Given Values

Start by substituting \( \vec{d}_{3} = 2\hat{i} + 4\hat{j} \) into the given equations.\( \vec{d}_{1} + \vec{d}_{2} = 5(2\hat{i} + 4\hat{j}) = 10\hat{i} + 20\hat{j} \)\( \vec{d}_{1} - \vec{d}_{2} = 3(2\hat{i} + 4\hat{j}) = 6\hat{i} + 12\hat{j} \)
02

Solve for \(\vec{d}_{1}\) and \(\vec{d}_{2}\)

Add the two equations to eliminate \(\vec{d}_{2}\):\ \(2\vec{d}_{1} = (10\hat{i} + 20\hat{j}) + (6\hat{i} + 12\hat{j})\ = 16\hat{i} + 32\hat{j} \)\ Therefore,\ \( \vec{d}_{1} = 8\hat{i} + 16\hat{j} \).
03

Find \(\vec{d}_{2}\)

Substitute \(\vec{d}_{1}\) back into one of the original equations (e.g., \(\vec{d}_{1} + \vec{d}_{2} = 10\hat{i} + 20\hat{j}\)):\ \( 8\hat{i} + 16\hat{j} + \vec{d}_{2} = 10\hat{i} + 20\hat{j} \)\ Solve for \(\vec{d}_{2}\):\ \( \vec{d}_{2} = (10\hat{i} + 20\hat{j}) - (8\hat{i} + 16\hat{j}) \)\( \vec{d}_{2} = 2\hat{i} + 4\hat{j} \)

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Key Concepts

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

Vector Components
In vector math, every vector can be split into its components. These components are the projections of the vector along the coordinate axes. For example, in a 2D plane, a vector \( \vec{A}\ \) can be expressed as \( \vec{A} = A_x \hat{i} + A_y \hat{j}\ \), where \( A_x\ \) and \( A_y\ \) are the components of the vector along the x and y axes, respectively.
Here, the unit vectors \(\hat{i}\ \) and \(\hat{j}\ \) represent the direction along the x-axis and y-axis, respectively.
By knowing the components, we can better understand and work with vectors, as this allows us to simplify and solve equations involving vectors.
Algebraic Manipulation
Algebraic manipulation is the process of rearranging and combining algebraic expressions to simplify or solve them. This often involves combining like terms, using the distributive property, and isolating variables.
In vector equations, algebraic manipulation can help in solving for unknown vectors by adding, subtracting, or factoring the vector components.
For example, in the given exercise, we start by substituting the components of \(\vec{d}_3\ \) into the equations, and then add or subtract the resulting vector equations to isolate and solve for \(\vec{d}_1\ \) and \(\vec{d}_2\ \).
It's crucial to keep track of each component separately and combine like terms correctly to avoid mistakes.
Vector Equations
A vector equation represents a relationship between two or more vectors. For instance, \(\vec{A} + \vec{B} = \vec{C}\ \) is an equation where vector \(\vec{A}\ \) and vector \(\vec{B}\ \) add up to result in vector \(\vec{C}\ \).
These equations can involve operations such as addition, subtraction, and scalar multiplication of vectors.
In the given problem, we are given two vector equations: \( \vec{d}_1 + \vec{d}_2 = 5 \vec{d}_3\ \) and \( \vec{d}_1 - \vec{d}_2 = 3 \vec{d}_3\ \).
By solving these vector equations, we can determine the unknown vectors \(\vec{d}_1\ \) and \(\vec{d}_2\ \).
This involves substituting the known vectors and using algebraic manipulation to isolate and solve for the desired vectors.

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Most popular questions from this chapter

A fire ant, searching for hot sauce in a picnic area, no goes through three displacements along level ground: \(\vec{d}_{1}\) for \(0.40 \mathrm{~m}\) southwest (that is, at \(45^{\circ}\) from directly south and from directly west), \(\vec{d}_{2}\) for \(0.50 \mathrm{~m}\) due east (that is, directly east), \(\vec{d}_{3}\) for \(0.60 \mathrm{~m}\) at \(60^{\circ}\) north of east (that is \(60.0^{\circ}\) toward the north from due east). Let the positive \(x\) direction be east and the positive \(y\) direction be north. What are (a) the \(x\) -component and (b) the \(y\) -component of \(\vec{d}_{1} ?\) What are (c) the \(x\) -component and (d) the \(y\) -component of \(\vec{d}_{2} ?\) What are (e) the \(x\) -component and (f) the \(y\) -component of \(\vec{d}_{3}\) ? What are \((\mathrm{g})\) the \(x\) -component, (h) the \(y\) -component, (i) the magnitude, and (j) the direction of the ant's net displacement? If the ant is to return directly to the starting point, \((\mathrm{k})\) how far and (I) in what direction should it move?

If \(\vec{B}\) is added to \(\vec{A}\), the result is \(6.0 \hat{\mathrm{i}}+1.0 \hat{\mathrm{j}}\). If \(\vec{B}\) is subtracted from \(\vec{A}\), the result is \(-4.0 \hat{\mathrm{i}}+7.0 \hat{\mathrm{j}}\). What is the magnitude of \(\vec{A}\) ?

Oasis \(B\) is \(25 \mathrm{~km}\) due east of oasis \(A\). Starting from oasis \(A\), a camel walks \(24 \mathrm{~km}\) in a direction \(15^{\circ}\) south of east and then walks \(8.0 \mathrm{~km}\) due north. How far is the camel then from oasis \(B\) ?

In a game of lawn chess, where pieces are moved between the centers of squares that are each \(1.00 \mathrm{~m}\) on edge, a knight is moved in the following way: (1) two squares forward, one square rightward; (2) two squares leftward, one square forward; (3) two squares forward, one square leftward. What are (a) the magnitude and (b) the angle (relative to "forward") of the knight's overall displacement for the series of three moves?

Consider how the components of a vector in the plane change if I change the reference point. Suppose I start with a coordinate system with an origin at \(O\). An arbitrary vector \(\vec{r}=x \hat{\mathrm{i}}+y \hat{\mathrm{j}}\) with coordinates \((x, y)\) specifies a point in this system. Suppose also that I have another point \(O^{\prime}\) specified in this coordinate system by a vector \(\vec{A}=A_{x} \hat{\mathrm{i}}+A_{y} \hat{\mathrm{j}}\). If I change my origin to \(O^{\prime}\) (without rotating the axes), what would the coordinates be for the point specified by \(\vec{r}\) ?
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