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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

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?
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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
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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.
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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
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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?
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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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