1. Home
  2. Precalculus
  3. Further applications of
  4. Vectors

8.8 Vectors  (Page 5/22)

<< Chapter < Page
  Precalculus     Page 5 / 22
Chapter >> Page >

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
Got questions? Get instant answers now!
Got questions? Get instant answers now!

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
Got questions? Get instant answers now!
Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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
Got questions? Get instant answers now!
Got questions? Get instant answers now!

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
Got questions? Get instant answers now!
Got questions? Get instant answers now!

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°

Got questions? Get instant answers now!

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

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
what is titration
John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
hello friend how are you
Muhammad Reply
fine, how about you?
Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
Reofrir Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
<< Chapter < Page Page > Chapter >>
Practice Key Terms 11

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Precalculus' conversation and receive update notifications?

Ask