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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

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Himanshu Reply
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
amani Reply
cosA\1+sinA=secA-tanA
Aasik Reply
why two x + seven is equal to nineteen.
Kingsley Reply
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
SAIDI
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simplify each radical by removing as many factors as possible (a) √75
Jason Reply
how is infinity bidder from undefined?
Karl Reply
what is the value of x in 4x-2+3
Vishal Reply
give the complete question
Shanky
4x=3-2 4x=1 x=1+4 x=5 5x
Olaiya
hi can you give another equation I'd like to solve it
Daniel
what is the value of x in 4x-2+3
Olaiya
if 4x-2+3 = 0 then 4x = 2-3 4x = -1 x = -(1÷4) is the answer.
Jacob
4x-2+3 4x=-3+2 4×=-1 4×/4=-1/4
LUTHO
then x=-1/4
LUTHO
4x-2+3 4x=-3+2 4x=-1 4x÷4=-1÷4 x=-1÷4
LUTHO
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was  1350  bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after  3  hours?
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litshani
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A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So, the length of the guy wire can be found by evaluating √(90000+160000). What is the length of the guy wire?
Maxwell Reply
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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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