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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

how can are find the domain and range of a relations
austin Reply
A cell phone company offers two plans for minutes. Plan A: $15 per month and $2 for every 300 texts. Plan B: $25 per month and $0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
Diddy Reply
6000
Robert
more than 6000
Robert
can I see the picture
Zairen Reply
How would you find if a radical function is one to one?
Peighton Reply
how to understand calculus?
Jenica Reply
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
rachel Reply
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
Reena Reply
what is foci?
Reena Reply
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations
Bryssen Reply
i want to sure my answer of the exercise
meena Reply
what is the diameter of(x-2)²+(y-3)²=25
Den Reply
how to solve the Identity ?
Barcenas Reply
what type of identity
Jeffrey
Confunction Identity
Barcenas
how to solve the sums
meena
hello guys
meena
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
Shakeena Reply
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
Rhudy Reply
what is a complex number used for?
Drew Reply
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
They are used in any profession where the phase of a waveform has to be accounted for in the calculations. Imagine two electrical signals in a wire that are out of phase by 90°. At some times they will interfere constructively, others destructively. Complex numbers simplify those equations
Tim

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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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