00:01
This problem, we're given the two vectors within in standard form.
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So vector u is xi plus 2j, and vector v is 7i minus 3j.
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We know that the angle between the two vectors is 30 degrees.
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What we want to do is we want to find the value for x.
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Well, we have a formula to help us find the angle between two vectors.
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It's the cosine of theta equals the dot product between the two vectors, so u.
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Dot v, divided by the magnitude of the vectors, so the magnitude of you times the magnitude of v.
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So let's go ahead and substitute in what we know.
00:32
Well, first off, we know the angle between the vectors is 30 degrees.
00:35
So we're going to have the cosine of 30 degrees equals.
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Well, now, for the numerator, we're going to have the dot product.
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Remember, that's the sum of the product of the x components to the y components.
00:46
So in this case, we're going to have 7 times x plus 2 times negative 3.
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Now, in the denominator, we have to find the magnitude of these vectors.
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Remember, to find the magnitude of a vector is the square root of the product of both components squared.
01:00
So the magnitude of u will be x squared plus 2 squared, and we'll multiply this by the magnitude of v, which would be 7 squared plus negative 3 squared.
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So now what we need to do is solve this equation for x.
01:13
Well, first off, we have the cosine of 30 degrees, which is root 3 over 2.
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Now in the numerator, we're going to have 7x, then we have 2 times negative 3, which is negative 6.
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Now in denominator, we have the square root of x squared plus 2 squared, which is 4.
01:26
So really, we just have the square root of x squared plus 4, getting multiplied by, well, 7 squared is 49, negative 3 squared is 9.
01:34
So when we add those, we get the square root of 58.
01:37
Okay, so now we just need to solve this equation somehow.
01:41
So the first thing i'm going to do is i'm going to cross multiply.
01:44
So in this case, we're going to have 2 times 7x, which is 14x, and then 2 times negative 6, which is negative 12.
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Now, going to go in the other directions, i'm going to multiply the square of the 3 by the square of the 58.
01:56
Well, 58 times 3 is 174, so we're going to have 107, or the square root of 174.
02:01
Times square root of x squared plus four.
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So now what we want to do to get rid of our radicals is we're going to square both sides of this equation.
02:10
So on the left -hand side, we have a binomial being squared.
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So we get a trinomial.
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The first term is the first term squared.
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Well, 14x squared is 196x squared.
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To get our second term, we multiplied the two together.
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So 14 times negative 12, and then we double it, which is equal to negative 36x.
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And for our last term, we square the last term, which is positive.
02:32
Of 144.
02:34
Now, on the right -hand side, we're squaring these two vectors...
