/*! 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 7 Let \(\vec{a}=2 \hat{i}-\hat{j}+... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 7

Let \(\vec{a}=2 \hat{i}-\hat{j}+\hat{k}, \vec{b}=\hat{i}+2 \hat{j}-\hat{k}, \vec{c}=\hat{i}+\hat{j}-2 \hat{k}\). A vector coplanar with \(\vec{b}\) and \(\vec{c}\), whose projection on \(\vec{a}\) is of magnitude \(\sqrt{\frac{2}{3}}\) is (1) \(2 \hat{i}+2 \hat{j}-3 \hat{k}\) (2) \(-2 \hat{i}-\hat{j}+5 \hat{k}\) (3) \(2 \hat{i}+3 \hat{j}+3 \hat{k}\) (4) \(2 \hat{i}+\hat{j}+5 \hat{k}\)

Short Answer

Expert verified
Correct option is (1): \((2\hat{i}+2\hat{j}-3\hat{k})\)

Step by step solution

01

Understand the problem

We need to find a vector coplanar with \(\backslashvec{b}\) and \(\backslashvec{c}\), whose projection on \(\backslashvec{a}\) is of magnitude \(\backslashsqrt\backslashfrac{2}{3}\).
02

General form of coplanar vector

A vector coplanar with \(\backslashvec{b}\) and \(\backslashvec{c}\) can be written as \(\backslashvec{r} = m\backslashvec{b} + n\backslashvec{c}\).
03

Projection of \(\backslashvec{r}\) on \(\backslashvec{a}\)

The projection of \(\backslashvec{r}\) on \(\backslashvec{a}\) is given by \[ \text{Proj}_{\backslashvec{a}}(\backslashvec{r}) = \frac{\backslashvec{r} \bullet \backslashvec{a}}{\backslashvec{a} \bullet \backslashvec{a}} \backslasha \backslashvec{a} \].
04

Calculate \(\backslashvec{a} \bullet \backslashvec{a}\)

Calculate \(\backslashvec{a} \bullet \backslashvec{a} = 2^2 + (-1)^2 + 1^2 = 4 + 1 + 1 = 6\).
05

Calculate \(\backslashvec{r} \bullet \backslashvec{a}\)

Let \(\backslashvec{r} = m\backslashvec{b} + n\backslashvec{c}\), then \(\backslashvec{r} \bullet \backslashvec{a} = (m \ (\hat{i} + 2\hat{j} - \hat{k}) + n \ (\hat{i} + \hat{j} - 2\hat{k})) \bullet (2 \hat{i} - \hat{j} + \hat{k})\).\~For simplicity, let \backslashvec{b} = \(b_1, b_2, b_3\)\~and \backslashvec{c} = \(c_1, c_2, c_3\), then: \(\backslashvec{r} \bullet \backslashvec{a} = 2m + (2m) \ (-1) + (-m) + 2n + n - 2n\).
06

Apply the magnitude condition

We are given that \(\text{Proj}_{\backslashvec{a}}(\backslashvec{r}) = \sqrt{\frac{2}{3}}\).\~Implying: \[ \left| \frac{\backslashvec{r} \bullet \backslashvec{a}}{6} \right| = \sqrt{\frac{2}{3}} \] \[ => \frac{\left|rm+n(3) \right|}{6} = \sqrt{\frac{2}{3}} \ => ||\vec{a} \cdot \vec{b}+\vec{c} \backslasha \backslashvec(t)|| = |\backslashsqrt(an)x (ans[solve]) = [mg/蓹(s)|\].
07

Compare options

Check each provided option for coplanarity with \(\backslashvec{b}\) and \(\backslashvec{c}\). Use step-6 equality to identify correct option.

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Key Concepts

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

Vector Projection
Vector projection is all about finding how much one vector 'lies' in the direction of another vector. To project vector \(\backslashvec{r}\) onto \(\backslashvec{a}\), we use the formula: \[\text{Proj}_{\backslashvec{a}}(\backslashvec{r}) = \frac{\backslashvec{r} \bullet \backslashvec{a}}{\backslashvec{a} \bullet \backslashvec{a}} \backslashvec{a} \] Here, \(\backslashvec{r} \bullet \backslashvec{a}\) is the dot product of \(\backslashvec{r}\) and \(\backslashvec{a}\), and \(\backslashvec{a} \bullet \backslashvec{a}\) is the dot product of \(\backslashvec{a}\) with itself. This method helps in breaking down complex three-dimensional problems into simpler components. In our exercise, the projection's magnitude on \(\backslashvec{a}\) is given, which means we need to use this value to find the exact vector. Knowing the projection is critical to solving many problems in physics and engineering.
Coplanar Vectors
Vectors are coplanar if they lie in the same plane. To check if vectors are coplanar, we can use the determinant method or the scalar triple product. In our problem, a vector coplanar with \(\backslashvec{b}\) and \(\backslashvec{c}\) is written as: \[\backslashvec{r} = m\backslashvec{b} + n\backslashvec{c} \] This formula means that \(\backslashvec{r}\) can be expressed as a linear combination of \(\backslashvec{b}\) and \(\backslashvec{c}\). Here, m and n are scalars. By using the condition of coplanarity, we narrow down potential candidates for our search, streamlining our problem-solving process significantly. For instance, while checking our options, the correct one should satisfy the coplanar condition with \(\backslashvec{b}\) and \(\backslashvec{c}\).
Dot Product
The dot product (or scalar product) is a way to multiply two vectors to get a scalar. It鈥檚 calculated as: \[\backslashvec{a} \bullet \backslashvec{b} = a_1b_1 + a_2b_2 + a_3b_3\] where \(\backslashvec{a} = (a_1, a_2, a_3)\) and \(\backslashvec{b} = (b_1, b_2, b_3)\). The dot product has many applications, such as calculating projections or checking orthogonality. In our exercise, the dot product is used to find the projection of \(\backslashvec{r}\) on \(\backslashvec{a}\). This calculation helps in determining the exact vector by comparing the known magnitude of the projection. For instance, the term \(\backslashvec{r} \bullet \backslashvec{a}\) needs to match a certain magnitude as per the given conditions, allowing us to solve for unknowns and identify the correct vector.

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Most popular questions from this chapter

The lines \(\frac{x-2}{1}=\frac{y-3}{1}=\frac{4-z}{k}\) and \(\frac{x-1}{k}=\frac{y-4}{2}=\frac{z-5}{1}\) are coplanar if \((1) k=0\) or \(-1\) \((2) \mathrm{k}=0\) or 1 (3) \(\mathrm{k}=0\) or \(-3\) (4) \(\mathrm{k}=3\) or \(-3\)

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