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Chapter 11: Problem 31

Let \(\mathbf{u}=\langle 4,-2\rangle, \mathbf{v}=\langle-4,6\rangle,\) and \(\mathbf{w}=\langle 0,8\rangle .\) Express the following vectors in the form \(\langle a, b\rangle\) $$\mathbf{w}-3 \mathbf{v}$$

Short Answer

Expert verified
Answer: The result of the vector expression \(\mathbf{w} - 3\mathbf{v}\) is the vector \(\langle 12, -10 \rangle\).

Step by step solution

01

Multiply vector \(\mathbf{v}\) by 3

To multiply the vector \(\mathbf{v}\) by the scalar 3, we multiply each component of \(\mathbf{v}\) by 3: \(3\mathbf{v} = \langle 3(-4), 3(6) \rangle = \langle -12, 18 \rangle\)
02

Subtract the resulting vector from \(\mathbf{w}\)

Now, we need to subtract the resulting vector \(3\mathbf{v} = \langle -12, 18 \rangle\) from the vector \(\mathbf{w} = \langle 0, 8 \rangle\). We can do this by subtracting the corresponding components of both vectors: \(\mathbf{w} - 3\mathbf{v} = \langle 0 - (-12), 8 - 18 \rangle = \langle 12, -10 \rangle\) The result of the expression \(\mathbf{w} - 3\mathbf{v}\) is the vector \(\langle 12, -10 \rangle\).

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Key Concepts

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

Scalar Multiplication
Scalar multiplication involves multiplying a vector by a single number, known as a scalar. This operation affects each component of a vector, scaling it up or down. Consider a vector \( \mathbf{v} = \langle x, y \rangle \). When we multiply \( \mathbf{v} \) by a scalar \( k \), we get a new vector \( k\mathbf{v} = \langle kx, ky \rangle \). This process changes the magnitude of the vector but keeps its direction the same unless the scalar is negative, in which case the direction is reversed.

In our example, multiplying \( \mathbf{v} = \langle -4, 6 \rangle \) by 3 gives us \( 3\mathbf{v} = \langle 3(-4), 3(6) \rangle = \langle -12, 18 \rangle \). Here, each component of the vector \( \mathbf{v} \) has been multiplied by 3.
Vector Subtraction
Vector subtraction involves finding the difference between two vectors by subtracting corresponding components. Given two vectors \( \mathbf{a} = \langle a_1, a_2 \rangle \) and \( \mathbf{b} = \langle b_1, b_2 \rangle \), the subtraction operation is done as follows: \( \mathbf{a} - \mathbf{b} = \langle a_1 - b_1, a_2 - b_2 \rangle \).

This operation is straightforward as it involves simple arithmetic subtraction for each pair of corresponding elements from the vectors.
  • Find the component-wise difference
  • Result is a new vector
In our problem, we needed to subtract \( 3\mathbf{v} = \langle -12, 18 \rangle \) from \( \mathbf{w} = \langle 0, 8 \rangle \), resulting in \( \mathbf{w} - 3\mathbf{v} = \langle 0 - (-12), 8 - 18 \rangle = \langle 12, -10 \rangle \).
Coordinate Representation
Vectors are often represented in coordinates as \( \langle a, b \rangle \), where \( a \) and \( b \) are the components along the x-axis and y-axis, respectively. This form makes vector calculations such as addition, subtraction, and scalar multiplication easier to visualize and compute.

For example, the vector \( \mathbf{w} = \langle 0, 8 \rangle \) indicates that its x-component is 0 and y-component is 8, meaning it lies directly along the y-axis.
  • Each vector component represents movement along one dimension
  • Vectors can be easily plotted on a graph
When expressing results like \( \langle 12, -10 \rangle \), we understand it as a vector with 12 units in the positive x-direction and 10 units in the negative y-direction, providing a clear visual representation of the vector's magnitude and direction.

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Most popular questions from this chapter

Consider the curve \(\mathbf{r}(t)=(a \cos t+b \sin t) \mathbf{i}+(c \cos t+d \sin t) \mathbf{j}+(e \cos t+f \sin t) \mathbf{k}\) where \(a, b, c, d, e,\) and \(f\) are real numbers. It can be shown that this curve lies in a plane. Find a general expression for a nonzero vector orthogonal to the plane containing the curve. $$\begin{aligned}\mathbf{r}(t)=&(a \cos t+b \sin t) \mathbf{i}+(c \cos t+d \sin t) \mathbf{j} \\ &+(e \cos t+f \sin t) \mathbf{k},\end{aligned}$$ where \(\langle a, c, e\rangle \times\langle b, d, f\rangle \neq \mathbf{0}\).

Evaluate the following definite integrals. $$\int_{0}^{\ln 2}\left(e^{t} \mathbf{i}+e^{t} \cos \left(\pi e^{t}\right) \mathbf{j}\right) d t$$

A golfer launches a tee shot down a horizontal fairway; it follows a path given by \(\mathbf{r}(t)=\left\langle a t,(75-0.1 a) t,-5 t^{2}+80 t\right\rangle,\) where \(t \geq 0\) measures time in seconds and \(\mathbf{r}\) has units of feet. The \(y\) -axis points straight down the fairway and the \(z\) -axis points vertically upward. The parameter \(a\) is the slice factor that determines how much the shot deviates from a straight path down the fairway. a. With no slice \((a=0),\) sketch and describe the shot. How far does the ball travel horizontally (the distance between the point the ball leaves the ground and the point where it first strikes the ground)? b. With a slice \((a=0.2),\) sketch and describe the shot. How far does the ball travel horizontally? c. How far does the ball travel horizontally with \(a=2.5 ?\)

Cauchy-Schwarz Inequality The definition \(\mathbf{u} \cdot \mathbf{v}=|\mathbf{u}||\mathbf{v}| \cos \theta\) implies that \(|\mathbf{u} \cdot \mathbf{v}| \leq|\mathbf{u}||\mathbf{v}|\) (because \(|\cos \theta| \leq 1\) ). This inequality, known as the Cauchy-Schwarz Inequality, holds in any number of dimensions and has many consequences. Triangle Inequality Consider the vectors \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{u}+\mathbf{v}\) (in any number of dimensions). Use the following steps to prove that \(|\mathbf{u}+\mathbf{v}| \leq|\mathbf{u}|+|\mathbf{v}|\) a. Show that \(|\mathbf{u}+\mathbf{v}|^{2}=(\mathbf{u}+\mathbf{v}) \cdot(\mathbf{u}+\mathbf{v})=|\mathbf{u}|^{2}+\) \(2 \mathbf{u} \cdot \mathbf{v}+|\mathbf{v}|^{2}\) b. Use the Cauchy-Schwarz Inequality to show that \(|\mathbf{u}+\mathbf{v}|^{2} \leq(|\mathbf{u}|+|\mathbf{v}|)^{2}\) c. Conclude that \(|\mathbf{u}+\mathbf{v}| \leq|\mathbf{u}|+|\mathbf{v}|\) d. Interpret the Triangle Inequality geometrically in \(\mathbb{R}^{2}\) or \(\mathbb{R}^{3}\).

Suppose \(\mathbf{u}\) and \(\mathbf{v}\) are vectors in the plane. a. Use the Triangle Rule for adding vectors to explain why \(|\mathbf{u}+\mathbf{v}| \leq|\mathbf{u}|+|\mathbf{v}| .\) This result is known as the Triangle Inequality. b. Under what conditions is \(|\mathbf{u}+\mathbf{v}|=|\mathbf{u}|+|\mathbf{v}| ?\)
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