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Chapter 1: Problem 74

Find the vector \(\vec{C}\) that satisfies the equation \(3 \hat{x}+6 \hat{y}\) \(10 \hat{z}+\vec{C}=-7 \hat{x}+14 \hat{y}\)

Short Answer

Expert verified
Answer: The vector \(\vec{C}\) that satisfies the given equation is \(\vec{C} = -10 \hat{x} + 8 \hat{y} - 10 \hat{z}\).

Step by step solution

01

Write the given equation with vectors in component form

The equation provided is: \(3 \hat{x} + 6 \hat{y} + 10 \hat{z} + \vec{C} = -7 \hat{x} + 14 \hat{y}\) We can rewrite this equation in component form as follows: \((3, 6, 10) + \vec{C} = (-7, 14, 0)\)
02

Isolate vector \(\vec{C}\)

In order to determine vector \(\vec{C}\), we need to isolate it in the equation. We will subtract the vector \((3, 6, 10)\) from both sides of the equation: \(\vec{C} = (-7, 14, 0) - (3, 6, 10)\)
03

Subtract the vectors

Now, subtract the corresponding components in the two vectors on the right-hand side of the equation: \(\vec{C} = (-7-3, 14-6, 0-10)\)
04

Simplify the result

Finally, simplify the numerical operations in the components: \(\vec{C} = (-10, 8, -10)\) So, the vector \(\vec{C}\) that satisfies the given equation is \(\vec{C} = -10 \hat{x} + 8 \hat{y} - 10 \hat{z}\).

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Most popular questions from this chapter

Advances in the field of nanotechnology have made it possible to construct chains of single metal atoms linked one to the next. Physicists are particularly interested in the ability of such chains to conduct electricity with little resistance. Estimate how many gold atoms would be required to make such a chain long enough to wear as a necklace. How many would be required to make a chain that encircled the Earth? If 1 mole of a substance is equivalent to roughly \(6.022 \cdot 10^{23}\) atoms, how many moles of gold are required for each necklace?

Roughly \(4.00 \%\) of what you exhale is carbon dioxide. Assume that \(22.4 \mathrm{~L}\) is the volume of \(1 \mathrm{~mole}\left(6.02 \cdot 10^{23} \mathrm{~mole}-\right.\) cules) of carbon dioxide and that you exhale \(0.500 \mathrm{~L}\) per breath. a) Estimate how many carbon dioxide molecules you breathe out each day. b) If each mole of carbon dioxide has a mass of \(44.0 \mathrm{~g}\), how many kilograms of carbon dioxide do you exhale in a year?

Two different forces, acting on the same object, are measured. One force is \(2.0031 \mathrm{~N}\) and the other force, in the same direction, is \(3.12 \mathrm{~N}\). These are the only forces acting on the object. Find the total force on the object to the correct number of significant figures.

If you draw a vector on a sheet of paper, how many components are required to describe it? How many components does a vector in real space have? How many components would a vector have in a four-dimensional world?

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