QA_9-2

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course Mth 277

qa 09_2

The magnitude of a vector `A = a_x `i + a_y `j in two-dimensional space is found using the Pythagorean Theorem to be sqrt(a_x^2 + a_y)^2.

The magnitude of a vector `A = a_1 `i + a_2 `j + a_3 `k, in 3-dimensional space, is found by two applications of the Pythagorean Theorem to be sqrt( a_1 ^ 2 + a_2 ^ 2 + a_3 ^2).

A unit vector is a vector whose magnitude is 1. If you divide any vector by its magnitude you get a unit vector, and its direction is the same as that of the original vector.

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Question: `q001. Write an equation expressing the following statement:

The distance between the point (x, y, z) and the point (3, -4, 2) is 5.

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Your solution:

First, we must use the distance formula

sqrt((x-3)^2+(y+4)^2+(z-2)^2)=5

To get rid of the square root, we need to square both sides

(x-3)^2+(y+4)^2+(z-2)^2)=25

confidence rating #$&*:

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Given Solution:

The distance between the point (x, y, z) and the point (3, -4, 2) is sqrt((3 - x)^2 + (-4 - y)^2 + (2 - z)^2) = sqrt( (x - 3)^2 + (y + 4)^2 + (z - 2)^2).

This distance is 5 if and only if

sqrt( (x - 3)^2 + (y + 4)^2 + (z - 2)^2). = 5.

Squaring both sides we obtain

(x - 3)^2 + (y + 4)^2 + (z - 2)^2) = 25.

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Self-critique (if necessary):OK

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Self-critique rating:OK

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Question: `q002. Using the equation from the preceding, find the value of y if we know that x = 2 and z = 1.

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Your solution:

Preceding equation is (x-3)^2+(y+4)^2+(z-2)^2)=25

(2-3)^2+(y+4)^2+(1-2)^2)=25

1+(y+4)^2+1=25

(y+4)^2=23

y+4=sqrt(23)

y=sqrt(23)-4

confidence rating #$&*:

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If x = 2 and z = 1 then our equation

(x - 3)^2 + (y + 4)^2 + (z - 2)^2) = 25

becomes

(2 - 3)^2 + (y + 4)^2 + (1 - 2)^2) = 25,

or

1 + (y+4)^2 + 1 = 25

so that

(y + 4)^2 = 23

and

(y + 4) = +- sqrt(23).

Solving for y we get

y = sqrt(23) - 4

or

y = -sqrt(23) - 4.

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Given Solution:

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Self-critique (if necessary):OK

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Self-critique rating:OK

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Question: `q003. Using the equation from the first question, substitute z = 0. The resulting equation describes a circle. What are its center and radius?

Answer the same questions if you substitute z = 1 rather than z = 0.

Answer the same questions if you substitute z = -1 rather than z = 0.

Answer the same questions if you substitute z = -4 rather than z = 0.

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Your solution:

Preceding equation is (x-3)^2+(y+4)^2+(z-2)^2)=25

With z=0, (x-3)^2+(y+4)^2+(0-2)^2)=25 (x-3)^2+(y+4)^2)=21 center (3,-4) radius=sqrt(21) The center stays constant since we are only changing z which will have an effect on the radius of the circle.

With z=1, (x-3)^2+(y+4)^2+(1-2)^2)=25 (x-3)^2+(y+4)^2)=24 center (3,-4) radius=sqrt(24)

With z=-1, (x-3)^2+(y+4)^2+(-1-2)^2)=25 (x-3)^2+(y+4)^2)=16 center (3,-4) radius=sqrt(16)=4

With z=-4, (x-3)^2+(y+4)^2+(-4-2)^2)=25 (x-3)^2+(y+4)^2)=-11 center (3,-4) radius=sqrt(-11) which is not possible in real numbers. Therefore, z=-4 does not exist on this graph.

confidence rating #$&*:

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Given Solution:

Our equation is

(x - 3)^2 + (y + 4)^2 + (z - 2)^2) = 25.

If z = 0 then our equation becomes

(x - 3)^2 + (y + 4)^2 + (0 - 2)^2) = 25,

which we easily rearrange to get

(x - 3)^2 + (y + 4)^2 = 21.

This is the equation of a circle in the x-y plane centered at (3, -4) and having radius sqrt(21), which is roughly 4.6.

If z = 1 then (z - 2)^2 becomes 1 and our equation becomes

(x - 3)^2 + (y + 4)^2 = 24.

This is the equation of a circle in the x-y plane centered at (3, -4) and having radius sqrt(24), which is roughly 4.9.

If z = -1 we get a circle of radius 6, centered at (3, -4).

If z = -4 we get the equation (x-3)^2 + (y+4)^2 = -11. Since squares can't be negative, there is no real-number solution for (x, y), and there will be no z = -4 point on this graph.

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Self-critique (if necessary):OK

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Self-critique rating:OK

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Question: `q004. Expand the equation you obtained in the first question by multiplying out the squares. Simplify into standard form, with all numbers and variable on the left and 0 on the right-hand side of the equation.

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Your solution:

(x-3)^2+(y+4)^2+(z-2)^2)=25

(x-3)(x-3)+(y+4)(y+4)+(z-2)(z-2)=25

x^2-6x+9+y^2+8y+16+z^2-4z+4=25

x^2-6x+y^2+8y+z^2-4z+4=0

confidence rating #$&*:

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Given Solution:

The equation

(x - 3)^2 + (y + 4)^2 + (z - 2)^2) = 25

becomes

x^2 - 6 x + 9 + y^2 + 8 y + 16 + z^2 - 4 z + 4 = 25

which in standard form is

x^2 - 6 x + y^2 + 8 y + z^2 - 4 z + 4 = 0.

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Self-critique (if necessary):OK

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Self-critique rating:OK

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Question: `q005. Is the vector 8 `i - 4 `j + 5 `k a multiple of the vector 4 `i + 2 `j - 5/2 `k?

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Your solution:

No, the two vectors are not multiples of eachother. This can be found out by looking at both equations closely.

The i components are a multiples of 2 of eachother while the j and k components can be seen to be multiples of -2 of eachother.

If all components had the same multiple then the vectors would be multiples of eachother. Since this is not the case, the two vectors are NOT multiples of eachother.

confidence rating #$&*:

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Given Solution: The vector 8 `i - 4 `j + 5 `k is a multiple of the vector 4 `i + 2 `j - 5/2 `k if, and only if, there exists a constant c such that

8 `i - 4 `j + 5 `k = c ( 4 `i + 2 `j - 5/2 `k ).

Setting the `i, `j and `k components of the left-hand side respectively equal to the same components of the right-hand side we get the three equations

8 = 4 c

-4 = 2 c

5 = -5/2 c

The first yields solution c = 2. The last two yield solution c = -2.

c can't take both values at the same time, so the equations are not simultaneously true.

It follows that neither vector is a multiple of the other.

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Self-critique (if necessary):

I should have solved for the multiples rather than saying what they were. Even though these were easy enough to do by looking at them, I still should have wrote them out for practice and to explain the concept properly.

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Self-critique rating:3

@&

Your explanation was fine, and equivalent to the given equations.

But the equations will be much more useful as you analyze various aspects of vectors.

*@

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Question: `q006. How does your answer to the preceding determine whether or not the two vectors are parallel?

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Your solution:

My answer determines that the vectors are not parallel. This is because for vectors to be parallel, they have to be multiples of eachother. If they are multiples of eachother then they are parallel, if not, then they are not parallel.

confidence rating #$&*:

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Given Solution: Two vectors are parallel if, and only if, one is a multiple of the other.

So the solution to the preceding shows that neither vector is a multiple of the other.

STUDENT SOLUTION (not quite complete): If a vector is a multiple of another, the two vectors are parallel.

Instructor Response:

Right idea but the statement you base the conclusion on needs to be stronger.

Had one vector been a multiple of the other, then your statement would allow us to conclude that the vectors are parallel.

However your statement doesn't address what happens if neither vector is a multiple of the other. That would require a statement equivalent to the following:

If two vectors are parallel, then each is a scalar multiple of the other.

We can combine the two statement into the single statement

Two vectors are scalar multiples of one another if and only if they are parallel.

This statement is equivalent to the statement in the given solution.

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Self-critique (if necessary):OK

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Self-critique rating:OK

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Question: `q007. What are the lengths of the sides of the triangle whose vertices are (4, 3 -2), (5, -1, 3) and (6, 4, 1)?

Sketch a triangle whose sides have these lengths, as best you can in a few minutes without meticulously measuring everything (i.e., try to keep the sides in approximately the right proportion). Based on your sketch does it seem plausible that this is a right triangle?

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Your solution:

Based on my sketch, I do not think it is plausible that this is a right triangle.

confidence rating #$&*:

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Given Solution:

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Self-critique (if necessary):OK

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Self-critique rating:OK"

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#*&!

&#This looks good. See my notes. Let me know if you have any questions. &#