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Chapter 11: Problem 1

Express the dot product of \(\mathbf{u}\) and \(\mathbf{v}\) in terms of their magnitudes and the angle between them.

Short Answer

Expert verified
Answer: The dot product of two vectors, u and v, is equal to the product of their magnitudes and the cosine of the angle between them, expressed as u · v = ||u|| * ||v|| * cos(θ).

Step by step solution

01

Recall the Dot Product Definition

The dot product of two vectors \(\mathbf{u}\) and \(\mathbf{v}\), denoted by \(\mathbf{u} \cdot \mathbf{v}\), is defined as the product of the magnitudes of the vectors and the cosine of the angle between them: \(\mathbf{u} \cdot \mathbf{v} = ||\mathbf{u}|| \, ||\mathbf{v}|| \cos{\theta}\) Where \(||\mathbf{u}||\) is the magnitude of vector \(\mathbf{u}\), \(||\mathbf{v}||\) is the magnitude of vector \(\mathbf{v}\), and \(\theta\) is the angle between the two vectors.
02

Solution

To express the dot product of \(\mathbf{u}\) and \(\mathbf{v}\) in terms of their magnitudes and the angle between them, we use the dot product definition from Step 1: \(\mathbf{u} \cdot \mathbf{v} = ||\mathbf{u}|| \, ||\mathbf{v}|| \cos{\theta}\) The dot product of vectors \(\mathbf{u}\) and \(\mathbf{v}\) is equal to the product of their magnitudes and the cosine of the angle between them.

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Most popular questions from this chapter

Properties of dot products Let \(\mathbf{u}=\left\langle u_{1}, u_{2}, u_{3}\right\rangle\) \(\mathbf{v}=\left\langle v_{1}, v_{2}, v_{3}\right\rangle,\) and \(\mathbf{w}=\left\langle w_{1}, w_{2}, w_{3}\right\rangle .\) Prove the following vector properties, where \(c\) is a scalar. $$c(\mathbf{u} \cdot \mathbf{v})=(c \mathbf{u}) \cdot \mathbf{v}=\mathbf{u} \cdot(c \mathbf{v})$$

Evaluate the following definite integrals. $$\int_{1 / 2}^{1}\left(\frac{3}{1+2 t} \mathbf{i}-\pi \csc ^{2}\left(\frac{\pi}{2} t\right) \mathbf{k}\right) d t$$

Orthogonal lines Recall that two lines \(y=m x+b\) and \(y=n x+c\) are orthogonal provided \(m n=-1\) (the slopes are negative reciprocals of each other). Prove that the condition \(m n=-1\) is equivalent to the orthogonality condition \(\mathbf{u} \cdot \mathbf{v}=0,\) where \(\mathbf{u}\) points in the direction of one line and \(\mathbf{v}\) points in the direction of the other line..

Show that two nonzero vectors \(\mathbf{u}=\left\langle u_{1}, u_{2}\right\rangle\) and \(\mathbf{v}=\left\langle v_{1}, v_{2}\right\rangle\) are perpendicular to each other if \(u_{1} v_{1}+u_{2} v_{2}=0\)

An object moves along a path given by \(\mathbf{r}(t)=\langle a \cos t+b \sin t, c \cos t+d \sin t\rangle, \quad\) for \(0 \leq t \leq 2 \pi\) a. What conditions on \(a, b, c,\) and \(d\) guarantee that the path is a circle? b. What conditions on \(a, b, c,\) and \(d\) guarantee that the path is an ellipse?
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