Vector components hard

In summary: And it's always good to check your answer.In summary, the conversation discusses two questions related to vectors and finding the vector component in a specific direction, as well as finding the parametric equations of a straight line of intersection between two planes. The individual asking the questions shares their approach to solving the problems and seeks confirmation on their solutions. Additionally, a suggestion is made to check the solution by plugging in the values and verifying that they satisfy the equations.
  • #1
dagg3r
67
0
hi all got some problems with vectors so waswondering if anyone can check what I've done
is correct thanks
question1.

find the vector component of 2i+j-k in the direction of i-3j+2k
basically i use the rule
a=(v*w^)*w^
w^= i-3j+2k / sqrt(14)

thus v=2i+j-k
therefore a=(2i+j-k)*( i-3j+2k / sqrt(14) ) * (i-3j+2k / sqrt(14))
a= [2(1)-3(1)-1(2) /sqrt(14)] * [i-3j+2k / sqrt(14)]
a= -3/sqrt(14)* i-3j+2k / sqrt(14)

thus the vector componentis a=-3/14( i-3j+2k) ?

2. find the parametric equationsof the straightline of intersection of the planes
x-2y+3z=5
3x+y-2z=1

i used gaussian elimination and got the tableu
1 -2 3 | 5
3 1 -2 | 1 R2-3R1

1 -2 3 | 5
0 7 -11| -14

x-2y+3z=5
7y-11z=-14
z=t

sub z=t into 7y-11z=-14
y=11t/7 - 2

sub y=11t/7 - 2 into x-2y+3z=5
x= 5 + 1/7t

therefore are the parametric equations
x= 5 + 1/7t
y=11t/7 - 2
z=t

??
thanks all
 
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  • #2
1. You found the projection vector along w, but it sounds like they just wanted the length of this vector. That is, just [itex]\vec v \cdot \hat w[/itex]. Think of it like this: if you rearranged your axes so that w pointed along the new x axis, what would be the new x component of v?

2. The easiest way to check this is to plug x, y, and z back into the two plane equations and see if they solve them for all t (ie, t drops out and you get something like 1=1).
 
  • #3


Hello! It looks like you have correctly found the vector component in the first question. Your steps and calculations are correct.

For the second question, your parametric equations are correct as well. Great job using Gaussian elimination to solve for the intersection of the two planes. Keep up the good work!
 

What are vector components?

Vector components are the individual parts that make up a vector. They are typically represented as x and y values in a two-dimensional vector and x, y, and z values in a three-dimensional vector.

What is the difference between magnitude and direction in vector components?

Magnitude refers to the size or length of a vector, while direction refers to the angle at which the vector is pointing. In vector components, magnitude is represented by the length of the vector and direction is represented by the x and y (or x, y, and z) values.

How do you calculate vector components?

To calculate vector components, you can use the Pythagorean theorem for two-dimensional vectors or the Pythagorean theorem and trigonometric functions for three-dimensional vectors. Alternatively, you can use vector addition to break down a vector into its components.

What are some common applications of vector components?

Vector components have many applications in physics and engineering, such as in the calculation of forces, velocities, and accelerations. They are also used in computer graphics and video game development to represent movement and animation.

How do you add and subtract vector components?

To add or subtract vector components, you simply add or subtract the corresponding x and y (or x, y, and z) values. This can be done using basic algebraic operations. Alternatively, you can use graphical methods such as vector addition or decomposition to calculate the resulting vector.

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