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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.

v = ( x 2 x 1 ) i + ( y 2 y 1 ) j = ( 6 2 ) i + ( 6 ( 6 ) ) j = 8 i + 12 j
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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.

v = ( x 2 x 1 ) i + ( y 2 y 1 ) j v = ( 2 ( 1 ) ) i + ( 7 3 ) j = 3 i + 4 j
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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.

Adding and subtracting vectors in rectangular coordinates

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

v + u = ( a + c ) i + ( b + d ) j v u = ( a c ) i + ( b d ) j

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

v 1 + v 2 = ( 2 + 4 ) i + ( 3 + 5 ) j = 6 i + 2 j
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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 .

Vector components in terms of magnitude and direction

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

cos θ = x | v | and sin θ = y | v | x = | v | cos θ y = | v | sin θ

Thus, v = x i + y j = | v | cos θ i + | v | sin θ j , and magnitude is expressed as | v | = 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

x = 7 cos ( 135° ) i = 7 2 2 y = 7 sin ( 135° ) j = 7 2 2

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

v = 7 2 2 i + 7 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 = 34 cos ( 59° ) i + 34 sin ( 59° ) j

Magnitude = 34

θ = tan 1 ( 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

bsc F. y algebra and trigonometry pepper 2
Aditi Reply
given that x= 3/5 find sin 3x
Adamu Reply
4
DB
remove any signs and collect terms of -2(8a-3b-c)
Joeval Reply
-16a+6b+2c
Will
is that a real answer
Joeval
(x2-2x+8)-4(x2-3x+5)
Ayush Reply
sorry
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
(X2-2X+8)-4(X2-3X+5)=0 ?
master
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master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
Y
master
X2-2X+8-4X2+12X-20=0 (X2-4X2)+(-2X+12X)+(-20+8)= 0 -3X2+10X-12=0 3X2-10X+12=0 Use quadratic formula To find the answer answer (5±Root11i)/3
master
Soo sorry (5±Root11* i)/3
master
x2-2x+8-4x2+12x-20 x2-4x2-2x+12x+8-20 -3x2+10x-12 now you can find the answer using quadratic
Mukhtar
explain and give four example of hyperbolic function
Lukman Reply
What is the correct rational algebraic expression of the given "a fraction whose denominator is 10 more than the numerator y?
Racelle Reply
y/y+10
Mr
Find nth derivative of eax sin (bx + c).
Anurag Reply
Find area common to the parabola y2 = 4ax and x2 = 4ay.
Anurag
A rectangular garden is 25ft wide. if its area is 1125ft, what is the length of the garden
Jhovie Reply
to find the length I divide the area by the wide wich means 1125ft/25ft=45
Miranda
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Jhovie
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Charmaine Reply
A banana.
Yaona
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Snalo Reply
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Mark Reply
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Miranda
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Miranda Drice
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jai
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Miranda
I am living in india
jai
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Miranda
what is the formula for calculating algebraic
Propessor Reply
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
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sita Reply
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Propessor
hi
Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
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jai
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jai
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Propessor
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jai
What is algebra
Pearl Reply
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
Miranda Drice would you mind teaching me mathematics? I think you are really good at math. I'm not good at it. In fact I hate it. 😅😅😅
Jeffrey
lolll who told you I'm good at it
Miranda
something seems to wispher me to my ear that u are good at it. lol
Jeffrey
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Miranda
but seriously, Im really bad at math. And I hate it. But you see, I downloaded this app two months ago hoping to master it.
Jeffrey
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Miranda
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Miranda
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Jeffrey
how about you? what grade are you now?
Jeffrey
I'm going to 11grade
Miranda
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Miranda
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Steve
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Steve
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Jeffrey
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Jeffrey
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Miranda
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Nelson Reply
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Miranda
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Jeffrey
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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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