Consider the following three vectors: a = 3i + 4j ; b = pi + 2j ; c = mi + nj , where
m, n and p are constants. If:
� a + b = 9i + 6j
� a.c = 10
� b x c = 2k
Show that p = 6, and determine values for m and n.
i got the p=6
as for the rest...what i did so far
b x c = 2k
so (6i x nj) + 2j x mi
6nk + 2mk = 2k...cancel out k's and simplify by dividing through by 2
so
3n + 1m = 1
___
a.c = 10
so
3i.mi + 4j.nj = 10
3m + 4n = 10
i would say solve simulatneously...but
when i do that, the answer doesn't come out to
m = 2 and n = 1
instead, i get m=3.2 and n=1.4
On the cross product.
ixj=k but jxi=-k. I think you missed the minus
The equation on the dot is correct.
wow...lol...look at what one little mistake can do
THANKS!~ i got it now
To solve for the values of m and n, let's start by simplifying the equations.
First, let's consider the equation b x c = 2k. We have:
(6i x nj) + (2j x mi) = 2k
Using the cross product of i and j (i x j = k), we can simplify this equation to:
6nj - 2mi = 2k
Comparing the k components, we get:
-2mi = 2k
Dividing both sides by 2, we have:
- mi = k
This implies that the constant m must be -1. Therefore, we have:
6nj - 2(-1) = 2k
6nj + 2 = 2k
Now let's move on to the equation a.c = 10. We have:
3m + 4n = 10
Solving these two equations simultaneously, we can substitute m = -1 into the equation:
6nj + 2 = 2k
Multiplying both sides by 3, we get:
18nj + 6 = 6k
Substituting this expression for 6nj + 2 into the equation 3m + 4n = 10, we have:
3*(-1) + 4n = 10
-3 + 4n = 10
Simplifying further:
4n = 13
n = 13/4
n ≈ 3.25
Substituting this value of n into the equation 3m + 4n = 10, we have:
3m + 4(3.25) = 10
3m + 13 = 10
3m = 10 - 13
3m = -3
m = -3/3
m = -1
Therefore, the values of m and n are m = -1 and n ≈ 3.25, respectively.
Now, let's calculate the value of p. We are given a + b = 9i + 6j. From the given vectors, a = 3i + 4j and b = pi + 2j. Therefore, we have:
(3i + 4j) + (pi + 2j) = 9i + 6j
Comparing the i components, we get:
3 + p = 9
p = 9 - 3
p = 6
Hence, we have shown that p = 6.