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Chapter 10: Q. 2 TF (page 826)

Explain why a single nonzero vector and a point uniquely determine a plane containing the point. (Hint: Think of the collection of vectors orthogonal to the given vector with the given point as the initial point of all of the vectors.)

Short Answer

Expert verified

A single nonzero vector and a point uniquely determine a plane containing the point if a given vector is a orthogonal to a plane.

Step by step solution

01

Step 1. Given Information

Explain why a single nonzero vector and a point uniquely determine a plane containing the point.

02

Step 2. A nonzero vector is one whose magnitude is greater than zero.

If a given vector is orthogonal to a plane, a single nonzero vector and a point uniquely identify a plane containing the point.

If two vectors are perpendicular to each other, they are orthogonal. The dot product of the two vectors, in other words, is zero.

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Most popular questions from this chapter

Find $u + v a n d u - v$ also sketch $u, v, u + v, u - v$

role="math" localid="1649603034674" $u = (1, - 4, 6) a n d v = (2, - 4, 7)$

Find $u + v$ and $u - v$ . Also sketch $u, v, u + v$ and $u - v$ .

role="math" localid="1649578020551" $u = (3, - 4) a n d v = (- 1, 5)$

Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: The sum formulas in Theorem 4.4 can be applied only to sums whose starting index value is $k = 1$ .

(b) True or False: $\sum_{k = 0}^{n} \frac{1}{k + 1} + \sum_{k = 1}^{n} k^{2}$ is equal to $\sum_{k = 0}^{n} \frac{k^{3} + k^{2} + 1}{k + 1}$ .

(c) True or False: $\sum_{k = 1}^{n} \frac{1}{k + 1} + \sum_{k = 0}^{n} k^{2}$ is equal to $\sum_{k = 1}^{n} \frac{k^{3} + k^{2} + 1}{k + 1}$ .

(d) True or False: $(\sum_{k = 1}^{n} \frac{1}{k + 1}) (\sum_{k = 1}^{n} k^{2})$ is equal to $\sum_{k = 1}^{n} \frac{k^{2}}{k + 1}$ .

(e) True or False: $\sum_{k = 0}^{m} \sqrt{k} + \sum_{k = m}^{n} \sqrt{k}$ is equal to $\sum_{k = 0}^{n} \sqrt{k}$ .

(f) True or False: $\sum_{k = 0}^{n} a_{k} = - a_{0} - a_{n} + \sum_{k = 1}^{n - 1} a_{k}$ .

(g) True or False: ${(\sum_{k = 1}^{10} a_{k})}^{2} = \sum_{k = 1}^{10} a_{k}^{2}$ .

(h) True or False: $\sum_{k = 1}^{n} {(e^{x})}^{2} = \frac{e^{x} (e^{x} + 1) (2 e^{x} + 1)}{6}$ .

Find a vector in the direction of $<3, 1, 2>$ and with magnitude 5.

Find a vector in the direction opposite to $<- 4, 5, - 1>$ and with magnitude 3.

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