- #1

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**a**and

**b**is cos^-1(4/21). Find p if

**a**= [6,3,-2] and

**b**= [-2, p, -4]

I did:

cos x= 4/21 =

**a**.

**b**/

**|a||b|**

The result comes out to be p=8/3 but it only satisfies that a dot b is 4 and not |a||b| = 21...What am I doing wrong?

- Thread starter PiRsq
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- #1

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I did:

cos x= 4/21 =

The result comes out to be p=8/3 but it only satisfies that a dot b is 4 and not |a||b| = 21...What am I doing wrong?

- #2

Hurkyl

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- #3

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cos theta = 3p-4/(7)root of p^2+20

I have no clue now of how to do it

- #4

Hurkyl

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(In case you don't remember, that is to multiply both sides by both denominators, thus going from a/b=c/d to a*d=b*c)

Then if only you knew an operation you can do to an equation to undo a square root...

- #5

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3p-4/root of p^2+20 = 4/21

Then I square the left side and cross multiply to get:

21(9p^2-24p+16)=4[7(p^2+20)]

am I right so far?

- #6

Hurkyl

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Incidentally, you should group your terms with parenthesis to make them more clear (and accurate): the LHS should be written something like

(3p-4)/( 7*sqrt(p^2+20) )

(where 'sqrt' stands for square root)

- #7

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(3p-4)/( 7*sqrt(p^2+20) )=4^2/21^2 then I end up getting numbers in the millions

Eventually I end up with an answer of 1.63 for p by using the quadratic formula, and my book says the answer is simply 4

- #8

Hurkyl

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Incidentally, one tip is to look for simplifications you can make at every step of the problem; for example, before doing anything, notice that the denominator of both sides is divisible by 7; you could multiply the equation through by 7 to cancel that out and reduce the size of the numbers with which you have to work.

- #9

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Ah yes, now I get the answer 4 and another negative number. But what is that negative number?

- #10

Hurkyl

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(2)^2 = 4 and (-2)^2 = 4

When you perform a noninvertable operation to an equation, it says "The solution(s) to the original equation is (are) among the solutions to this new equation". Generally it's good practice to check your solutions when you get them, but it becomes a necessity when you use noninvertable operations like squaring.

In many types of problems where there squaring introduces "false" solutions, the false solutions correspond to some sort of reversal of sign or direction. In this particular case, it corresponds to the case when cos θ = -4/21

- #11

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Ok great, thanks Hurkyl!

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