8.8 Vectors (Page 5/22)

Precalculus

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Writing a vector in terms of i And j

Given a vector $v$ with initial point $P = (2, −6)$ and terminal point $Q = (−6, 6),$ write the vector in terms of $i$ and $j .$

Begin by writing the general form of the vector. Then replace the coordinates with the given values.

\begin{array}{l} v = (x_{2} - x_{1}) i + (y_{2} - y_{1}) j \\ = (- 6 - 2) i + (6 - (- 6)) j \\ = - 8 i + 12 j \end{array}

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Writing a vector in terms of i And j Using initial and terminal points

Given initial point $P_{1} = (- 1, 3)$ and terminal point $P_{2} = (2, 7),$ write the vector $v$ in terms of $i$ and $j .$

Begin by writing the general form of the vector. Then replace the coordinates with the given values.

\begin{array}{l} v = (x_{2} - x_{1}) i + (y_{2} - y_{1}) j \\ v = (2 - (- 1)) i + (7 - 3) j \\ = 3 i + 4 j \end{array}

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Write the vector $u$ with initial point $P = (- 1, 6)$ and terminal point $Q = (7, - 5)$ in terms of $i$ and $j .$

$u = 8 i - 11 j$

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Performing operations on vectors in terms of i And j

When vectors are written in terms of $i$ and $j,$ we can carry out addition, subtraction, and scalar multiplication by performing operations on corresponding components.

a general note label

Adding and subtracting vectors in rectangular coordinates

Given v = a i + b j and u = c i + d j , then

\begin{matrix} v + u = (a + c) i + (b + d) j \\ v - u = (a - c) i + (b - d) j \end{matrix}

Finding the sum of the vectors

Find the sum of $v_{1} = 2 i - 3 j$ and $v_{2} = 4 i + 5 j .$

According to the formula, we have

\begin{array}{l} v_{1} + v_{2} = (2 + 4) i + (- 3 + 5) j \\ = 6 i + 2 j \end{array}

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Calculating the component form of a vector: direction

We have seen how to draw vectors according to their initial and terminal points and how to find the position vector. We have also examined notation for vectors drawn specifically in the Cartesian coordinate plane using $i and j .$ For any of these vectors, we can calculate the magnitude. Now, we want to combine the key points, and look further at the ideas of magnitude and direction.

Calculating direction follows the same straightforward process we used for polar coordinates. We find the direction of the vector by finding the angle to the horizontal. We do this by using the basic trigonometric identities, but with $| v |$ replacing $r .$

a general note label

Vector components in terms of magnitude and direction

Given a position vector $v = ⟨ x, y ⟩$ and a direction angle $θ,$

\begin{array}{l} \cos θ = \frac{x}{| v |} & and \begin{matrix}  \end{matrix} & \sin θ = \frac{y}{| v |} \\ x = | v | \cos θ \begin{matrix}  \end{matrix} & y = | v | \sin θ \end{array}

Thus, $v = x i + y j = | v | \cos θ i + | v | \sin θ j,$ and magnitude is expressed as $| v | = \sqrt{x^{2} + y^{2}} .$

Writing a vector in terms of magnitude and direction

Write a vector with length 7 at an angle of 135° to the positive x -axis in terms of magnitude and direction.

Using the conversion formulas $x = | v | \cos θ i$ and $y = | v | \sin θ j,$ we find that

\begin{array}{l} x = 7 \cos (135°) i \\ = - \frac{7 \sqrt{2}}{2} \\ y = 7 \sin (135°) j \\ = \frac{7 \sqrt{2}}{2} \end{array}

This vector can be written as $v = 7 \cos (135°) i + 7 \sin (135°) j$ or simplified as

v = - \frac{7 \sqrt{2}}{2} i + \frac{7 \sqrt{2}}{2} j

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A vector travels from the origin to the point $(3, 5) .$ Write the vector in terms of magnitude and direction.

$v = \sqrt{34} \cos (59°) i + \sqrt{34} \sin (59°) j$

Magnitude = $\sqrt{34}$

$θ = \tan^{- 1} (\frac{5}{3}) = 59.04°$

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Finding the dot product of two vectors

As we discussed earlier in the section, scalar multiplication involves multiplying a vector by a scalar, and the result is a vector. As we have seen, multiplying a vector by a number is called scalar multiplication. If we multiply a vector by a vector, there are two possibilities: the dot product and the cross product . We will only examine the dot product here; you may encounter the cross product in more advanced mathematics courses.

Questions & Answers

Ayele, K., 2003. Introductory Economics, 3rd ed., Addis Ababa.

Widad Reply

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Ariel

What is economics

Widad Reply

the study of how humans make choices under conditions of scarcity

AI-Robot

U(x,y) = (x×y)1/2 find mu of x for y

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U(x,y) = (xÃy)1/2 find mu of x for y

Desalegn

what is ecnomics

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this is the study of how the society manages it's scarce resources

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macroeconomic is the branch of economics which studies actions, scale, activities and behaviour of the aggregate economy as a whole.

husaini

etc

husaini

difference between firm and industry

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what's the difference between a firm and an industry

Abdul

firm is the unit which transform inputs to output where as industry contain combination of firms with similar production 😅😅

Abdulraufu

Suppose the demand function that a firm faces shifted from Qd  120 3P to Qd  90  3P and the supply function has shifted from QS  20  2P to QS 10  2P . a) Find the effect of this change on price and quantity. b) Which of the changes in demand and supply is higher?

Toofiq Reply

explain standard reason why economic is a science

innocent Reply

factors influencing supply

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what is economic.

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scares means__________________ends resources. unlimited

Jan

economics is a science that studies human behaviour as a relationship b/w ends and scares means which have alternative uses

Jan

calculate the profit maximizing for demand and supply

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Why qualify 28 supplies

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out-of-pocket costs for a firm, for example, payments for wages and salaries, rent, or materials

AI-Robot

concepts of supply in microeconomics

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economic overview notes

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identify a demand and a supply curve

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i don't know

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there's a difference

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Demand curve shows that how supply and others conditions affect on demand of a particular thing and what percent demand increase whith increase of supply of goods

Israr

Hi Sir please how do u calculate Cross elastic demand and income elastic demand?

Abari

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Source: OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6

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