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In Exercises $15 - 18$ a. Find the area of the triangle determined by the points $P , Q ,$ and $R .$ b. Find a unit vector perpendicular to plane $P Q R$ . $$ P ( 1,1,1 ) , \quad Q ( 2,1,3 ) , \quad R ( 3 , - 1,1 ) $$

In Exercises $15 - 18$ a. Find the area of the triangle determined by the points $P , Q ,$ and $R .$ b. Find a unit vector perpendicular to plane $P Q R$ . $$ P ( 1,1,1 ) , \quad Q ( 2,1,3 ) , \quad R ( 3 , - 1,1 ) $$

Thomas Calculus

Vectors and the Geometry of Space

The Cross Product
Find $r$ so that the vector from the point $A(1,-1,3)$ to the point $B(3,0,5)$ is orthogonal to the vector from $A$ to the point $P(r, r, r)$

Find $r$ so that the vector from the point $A(1,-1,3)$ to the point $B(3,0,5)$ is orthogonal to the vector from $A$ to the point $P(r, r, r)$

Calculus Early Transcendentals

THREE-DIMENSIONAL SPACE; VECTORS

Dot Product; Projections

For each given $p$ -series, identify $p$ and determine whether the series converges. $$ \text { (a) } \sum_{k=1}^{\infty} k^{-4 / 3} \text { (b) } \sum_{k=1}^{\infty} \frac{1}{\sqrt[4]{k}} \quad \text { (c) } \sum_{k=1}^{\infty} \frac{1}{\sqrt[3]{k^{5}}} \quad \text { (d) } \sum_{k=1}^{\infty} \frac{1}{k^{\pi}} $$

For each given $p$ -series, identify $p$ and determine whether the series converges. $$ \text { (a) } \sum_{k=1}^{\infty} k^{-4 / 3} \text { (b) } \sum_{k=1}^{\infty} \frac{1}{\sqrt[4]{k}} \quad \text { (c) } \sum_{k=1}^{\infty} \frac{1}{\sqrt[3]{k^{5}}} \quad \text { (d) } \sum_{k=1}^{\infty} \frac{1}{k^{\pi}} $$

Calculus Early Transcendentals

INFINITE SERIES

Convergence Tests

For each given $p$ -series, identify $p$ and determine whether the series converges. $$ \text { (a) } \sum_{k=1}^{\infty} \frac{1}{k^{3}} \quad \text { (b) } \sum_{k=1}^{\infty} \frac{1}{\sqrt{k}} \quad \text { (c) } \sum_{k=1}^{\infty} k^{-1} \quad \text { (d) } \sum_{k=1}^{\infty} k^{-2 / 3} $$

For each given $p$ -series, identify $p$ and determine whether the series converges. $$ \text { (a) } \sum_{k=1}^{\infty} \frac{1}{k^{3}} \quad \text { (b) } \sum_{k=1}^{\infty} \frac{1}{\sqrt{k}} \quad \text { (c) } \sum_{k=1}^{\infty} k^{-1} \quad \text { (d) } \sum_{k=1}^{\infty} k^{-2 / 3} $$

Calculus Early Transcendentals

INFINITE SERIES

Convergence Tests

Questions asked

INSTANT ANSWER

Find the vector component of \( \mathbf{v}=\langle 4,-1,7\rangle \) along \( \mathbf{b}=\langle 1,4,-8\rangle \) and the vector component of \( \mathbf{v} \) orthogonal to \( \mathbf{b} \). Enter the exact answers. The vector component of \( \mathbf{v} \) along \( \mathbf{b} \) is \langle \rangle. The vector component of \( \mathbf{v} \) orthogonal to \( \mathbf{b} \) is \langle

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INSTANT ANSWER

Consider the following equilibrium system: \[ \mathrm{N}_{2} \mathrm{O}_{4}(\mathrm{~g}) \leftrightarrows 2 \mathrm{NO}_{2}(\mathrm{~g}) \] At a particular time, the rate of \( \mathrm{N}_{2} \mathrm{O}_{4}(\mathrm{~g}) \) being formed is \( 1.0 \mathrm{M} / \mathrm{s} \), while the rate of \( \mathrm{NO}_{2}(\mathrm{~g}) \) being formed is \( 4.0 \mathrm{M} / \mathrm{s} \). Is this system at dynamic equilibrium? Yes, the rate of the forward reaction is equal to the rate of the reverse reaction. No, the rate of the forward reaction is greater than the rate of the reverse reaction. No, the rate of the forward reaction is less than the rate of the reverse reaction. The information provided is not sufficient to determine equilibrium position.

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ANSWERED

Supreeta N verified

Numerade educator

Find the lengths of the sides of the triangle with the indicated vertices, and determine whether the triangle is a right triangle, an isosceles triangle, or neither. A(4, -1, -1), B(2, 0, -4), C(3, 5, -1) |AB| = |AC| = |BC| = right triangle isosceles triangle neither

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ANSWERED

Supreeta N verified

Numerade educator

Find u × v, v × u, and v × v. u = 3i + 7k v = 5i + 9j ? 2k. (a) u × v (b) v × u (c) v × v

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Melissa Munoz verified

Numerade educator

Find sets of parametric equations and symmetric equations of the line that passes through the given point and is parallel to the given vector or line. (For each line, write the direction numbers as integers.) Point Parallel to (-9, 0, 2) v = 8i + 4j - 6k (a) parametric equations (Enter your answers as a comma-separated list.) (b) symmetric equations x/8 = y = z/6 (x + 9)/8 = y/4 = (2 - z)/6 8x = y/4 = 6z (x - 9)/8 = y = z/6

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ANSWERED

Supreeta N verified

Numerade educator

For the reaction, A ? B + C, the rate law is rate = k[A]. If it takes 80.0 seconds for 70.0% of a 10.0 gram sample of A to be transformed into products, what is the value of the rate constant? 0.00450 s?¹ 0.0290 s?¹ 0.00530 s?¹ 5.40 s?¹ 0.0150 s?¹

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ANSWERED

Supreeta N verified

Numerade educator

Consider the following balanced equation: 4 NH3 + 7 O2 ? 6 H2O + 4 NO2 What is the order with respect to ammonia? -1 0 1/2 1 3/2 2 3 4 cannot be determined

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ANSWERED

Supreeta N verified

Numerade educator

Consider the following balanced equation: 4 NH3 + 7 O2 ? 6 H2O + 4 NO2 What is the order with respect to ammonia? -1 0 1/2 1 3/2 2 3 4 cannot be determined

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ANSWERED

Supreeta N verified

Numerade educator

For a first-order reaction with a single reactant, after 230.0 seconds, 10.0% of the reactant remains. The rate constant for the reaction is therefore 0.0100 s?¹ 0.0510 s?¹ 100 s?¹ 0.000640 s?¹ 0.0915 s?¹

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ANSWERED

Supreeta N verified

Numerade educator

At a certain temperature the following data was collected for the reaction: 2NO2(g) + H2(g) ? 2HNO2(g) [NO2](M) | [H2](M) | Initial rate (M/s) 0.00100 | 0.00400 | 0.00200 0.00200 | 0.00400 | 0.00800 0.00300 | 0.00400 | 0.0180 0.00400 | 0.00100 | 0.00800 0.00400 | 0.00200 | 0.0160 0.00400 | 0.00300 | 0.0240 What is the rate law for this reaction? rate = k[NO2]2[H2]^2 rate = k[NO2]^0.5[H2] rate = k[NO2]^2[H2]^3 rate = k[NO2]^2[H2] rate = k[NO2]^2[H2]^2

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