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Chapter 9: Problem 16

Given that \(\lim _{t \rightarrow a} r_{1}(t)=\mathbf{i}-2 \mathbf{j}+\mathbf{k}\) and \(\lim _{t \rightarrow a} \mathbf{r}_{2}(t)=2 \mathbf{i}+\) \(5 \mathbf{j}+7 \mathbf{k}\), find: (a) \(\lim _{t \rightarrow a}\left[-4 \mathbf{r}_{1}(t)+3 \mathbf{r}_{2}(t)\right]\) (b) \(\lim _{t \rightarrow a} \mathbf{r}_{1}(t) \cdot \mathbf{r}_{2}(t) .\)

Short Answer

Expert verified
(a) \(2\mathbf{i} + 23\mathbf{j} + 17\mathbf{k}\); (b) \(-1\).

Step by step solution

01

Identify the given limits

We are given the following limits: \( \lim _{t \rightarrow a} \mathbf{r}_{1}(t) = \mathbf{i} - 2\mathbf{j} + \mathbf{k}\) \( \lim _{t \rightarrow a} \mathbf{r}_{2}(t) = 2\mathbf{i} + 5\mathbf{j} + 7\mathbf{k}\)
02

Apply the limit rules for vectors to part (a)

Using the linearity of limits, we can find the limit of the expression by evaluating each component separately:\[\lim _{t \rightarrow a}\left[-4 \mathbf{r}_{1}(t)+3 \mathbf{r}_{2}(t)\right] = -4 \lim _{t \rightarrow a} \mathbf{r}_{1}(t) + 3 \lim _{t \rightarrow a} \mathbf{r}_{2}(t)\]
03

Substitute the given limits into the expression for part (a)

Substitute the limits from step 1 into the expression:\[= -4(\mathbf{i} - 2\mathbf{j} + \mathbf{k}) + 3(2\mathbf{i} + 5\mathbf{j} + 7\mathbf{k})\]Evaluate this expression component-wise:\[-4(\mathbf{i}) = -4\mathbf{i}\] \[-4(-2\mathbf{j}) = 8\mathbf{j}\] \[-4(\mathbf{k}) = -4\mathbf{k}\]\[+3(2\mathbf{i}) = 6\mathbf{i}\]\[+3(5\mathbf{j}) = 15\mathbf{j}\]\[+3(7\mathbf{k}) = 21\mathbf{k}\]Add these results together:\[(-4\mathbf{i} + 6\mathbf{i}) + (8\mathbf{j} + 15\mathbf{j}) + (-4\mathbf{k} + 21\mathbf{k})\]Which simplifies to:\[2\mathbf{i} + 23\mathbf{j} + 17\mathbf{k}\]
04

Compute the dot product for part (b)

To find the dot product, use the components of the given limits:\[\lim _{t \rightarrow a} \mathbf{r}_{1}(t) \cdot \mathbf{r}_{2}(t) = (\mathbf{i} - 2\mathbf{j} + \mathbf{k}) \cdot (2\mathbf{i} + 5\mathbf{j} + 7\mathbf{k})\]The dot product is calculated as follows:\[= (1 \cdot 2) + (-2 \cdot 5) + (1 \cdot 7)\]Calculate each product:\(1 \cdot 2 = 2\), \(-2 \cdot 5 = -10\), \(1 \cdot 7 = 7\)Sum these values:\(2 - 10 + 7 = -1\).

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Key Concepts

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

Limits of Vector Functions
In calculus, approaching the limit of a vector function is similar to handling limits of regular functions but applied to each vector component separately. Specifically, if you have two vector functions \( \mathbf{r}_1(t) \) and \( \mathbf{r}_2(t) \), their limits as \( t \) approaches a specific value \( a \) are given component-wise.
  • Each component of a vector is evaluated with respect to \( t \, \rightarrow \, a \).
  • In the exercise, we have \( \lim _{t \, \rightarrow \, a} \mathbf{r}_1(t) = \mathbf{i} - 2\mathbf{j} + \mathbf{k} \) and \( \lim _{t \, \rightarrow \, a} \mathbf{r}_2(t) = 2 \mathbf{i} + 5 \mathbf{j} + 7 \mathbf{k} \).
  • When calculating the limit for a vector operation such as \(-4 \mathbf{r}_1(t) + 3 \mathbf{r}_2(t)\), apply the limits like a regular function, component by component.
This means you handle the expression in a linear fashion, making use of the properties of limits to simplify it effectively and extract results component-wise.
Dot Product Calculation
The dot product, also known as the scalar product, is an essential operation in vector calculus that yields a scalar value from two vectors. Understanding how to compute the dot product involves multiplying corresponding components of the vectors and then summing up these products.
  • For the vectors in the exercise, \( \mathbf{r}_1(t) = \mathbf{i} - 2\mathbf{j} + \mathbf{k} \) and \( \mathbf{r}_2(t) = 2 \mathbf{i} + 5 \mathbf{j} + 7 \mathbf{k} \), calculate each of the products first:
  • The dot product formula is \( a_1 \times b_1 + a_2 \times b_2 + a_3 \times b_3 \), where \( a \) and \( b \) represent the components of the vectors.
  • Therefore, \( (1 \times 2) + (-2 \times 5) + (1 \times 7) \). These give us 2, -10, and 7, respectively.
  • Adding these gives the result: \( 2 - 10 + 7 = -1 \).
By thoroughly understanding each step, you ensure that the dot product correctly simplifies to a single scalar, which is critical for solving many problems in physics and engineering.
Vector Operations
Vector operations encompass a range of mathematical manipulations of vectors. Two fundamental operations include addition and scalar multiplication. These are crucial to understanding complex systems represented by vectors.
  • Scalar Multiplication: Involves multiplying each component of the vector by a scalar (constant). For instance, \(-4 \mathbf{r}_1(t)\) leads to multiplying each part of \( \mathbf{r}_1(t)\) by -4.
  • Vector Addition: In vector addition, you simply add the corresponding components of the vectors. This can be observed when we calculate \(-4 \mathbf{r}_1(t) + 3 \mathbf{r}_2(t)\).
Apply these operations carefully:
  • For scalar multiplication, ensure that each dimension is multiplied by the scalar.
  • When adding the vectors, sum up the same components from each vector together. For example: for \( \mathbf{i} \) terms, \(-4\mathbf{i} + 6\mathbf{i}\) gives \(2\mathbf{i}\).
These operations help in transforming and combining vectors, giving us new insights into various vector-based scenarios.

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Most popular questions from this chapter

In Problems, evaluate the given iterated integral by changing to polar coordinates. $$ \int_{-\sqrt{\pi}}^{\sqrt{\pi}} \int_{0}^{\sqrt{\pi-x^{2}}} \sin \left(x^{2}+y^{2}\right) d y d x $$

In Problems \(7-10\), find the Jacobian of the transformation \(T\) from the \(u v\) -plane to the \(x y\) -plane. $$ u=\frac{2 x}{x^{2}+y^{2}}, v=\frac{-2 y}{x^{2}+y^{2}} $$

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