(a) Given that A= 3i - j + 2k and
B=2i + 3j - k find A x B
(b) if A= i + j, B= 2i - 3j + k and
C= 4j - 3k
find (i) (AxB)x C
(ii) Ax(BxC)
I will do one. These are much easier to do as matrix multipication on your calc, I would learn to do that.
1)
AxB= (3i-j+2k)x(2i+3j-k)
= 3*3ixj-3ixk-2jxi+jxk+4kxi+6kxj
well, ixj=k
jxk=i
kxi=j (and the opposite order means -)
AxB= 9k+3j+2k+i+4j-6i
check that, and combine terms.
(a) To find the cross product of two vectors A and B, we can use the following formula:
A x B = (AyBz - AzBy)i - (AxBz - AzBx)j + (AxBy - AyBx)k
Given A = 3i - j + 2k and B = 2i + 3j - k, we can substitute these values into the formula:
Ax = 3
Ay = -1
Az = 2
Bx = 2
By = 3
Bz = -1
Using the formula for cross product, we can calculate:
(AyBz - AzBy)i = (-1 * -1 - 2 * 3)i = (-1 + 6)i = 5i
(AxBz - AzBx)j = (3 * -1 - 2 * 2)j = (-3 - 4)j = -7j
(AxBy - AyBx)k = (3 * 3 - (-1) * 2)k = (9 + 2)k = 11k
Combining these terms, we get:
A x B = 5i - 7j + 11k
Therefore, the cross product of A and B is 5i - 7j + 11k.
(b) (i) To find (A x B) x C, we can first calculate the cross product of A and B, and then take the cross product of the result with C.
Given A = i + j, B = 2i - 3j + k, and C = 4j - 3k, we can substitute these values into the formula:
(A x B) x C = ((A x B)yCz - (A x B)zCy)i - ((A x B)xCz - (A x B)zCx)j + ((A x B)xBz - (A x B)zBx)k
To calculate each term, we need to find the cross product of A and B first. Using the formula from part (a):
(A x B) = (AyBz - AzBy)i - (AxBz - AzBx)j + (AxBy - AyBx)k
Ax = 1
Ay = 1
Az = 0
Bx = 2
By = -3
Bz = 1
(A x B) = (1 * 1 - 0 * -3)i - (1 * 1 - 0 * 2)j + (1 * -3 - 1 * 2)k
= (1 + 0)i - (1 + 0)j + (-3 - 2)k
= i - j - 5k
Now, we can substitute this into our formula:
((A x B)yCz - (A x B)zCy)i - ((A x B)xCz - (A x B)zCx)j + ((A x B)xBz - (A x B)zBx)k
((i - j - 5k)yCz - (i - j - 5k)zCy)i - ((i - j - 5k)xCz - (i - j - 5k)zCx)j + ((i - j - 5k)xBz - (i - j - 5k)zBx)k
To simplify this expression, we can distribute and substitute the values of C:
((1 - 1 - 5 * 0) * 4 * -3 - (1 - 1 - 5 * 0) * -3 * 4)i - ((1 - 1 - 5 * 0) * 0 * -3 - (1 - 1 - 5 * 0) * 0 * 4)j + ((1 - 1 - 5 * 0) * 2 * 4 - (1 - 1 - 5 * 0) * 4 * 2)k
(0 * -12 - 0 * -12)i - (0 * -3 - 0 * 4)j + (0 * 8 - 0 * 8)k
0i - 0j + 0k
Therefore, (A x B) x C evaluates to 0i - 0j + 0k, which is the zero vector.
(ii) To find A x (B x C), we can first calculate the cross product of B and C, and then take the cross product of A with the result.
Given A = i + j, B = 2i - 3j + k, and C = 4j - 3k, we can substitute these values into the formula:
B x C = (ByCz - BzCy)i - (BxCz - BzCx)j + (BxBz - ByBx)k
Bx = 2
By = -3
Bz = 1
Cx = 0
Cy = 4
Cz = -3
(B x C) = (-3 * -3 - 1 * 4)i - (2 * -3 - 1 * 0)j + (2 * 4 - (-3) * 0)k
= (9 + 4)i - (-6 - 0)j + (8 - 0)k
= 13i + 6j + 8k
Now, we can substitute this into the formula:
A x (B x C) = (Ay(B x C)z - Az(B x C)y)i - (AxB x Cz - Az(B x C)x)j + (AxB x Cy - Ay(B x C)x)k
(A x (B x C)) = (1 * 6 - 0 * 13)i - (1 * 8 - 0 * 13)j + (1 * 13 - 1 * 6)k
= (6)i - (8)j + (13 - 6)k
= 6i - 8j + 7k
Therefore, A x (B x C) evaluates to 6i - 8j + 7k.
To find the cross product of two vectors, you can use the determinant method or the component method. I will explain both methods.
(a) Using the determinant method:
To find the cross product of A and B using the determinant method, you need to arrange the components of A and B in a determinant matrix format and then evaluate the determinant.
A = 3i - j + 2k
B = 2i + 3j - k
Arrange the components in determinant format:
| i j k |
| 3 -1 2 |
A x B =| 2 3 -1 |
Evaluate the determinant using the formula:
(A x B) = i(3 * (-1) - 2 * 3) - j(2 * (-1) - 2 * 3) + k(2 * 3 - 3 * (-1))
(A x B) = -9i - 3j + 9k
So, A x B = -9i - 3j + 9k.
(b) Using the component method:
To find the cross product of A and B using the component method, you can use the following formula:
(A x B) = (AyBz - AzBy)i - (AxBz - AzBx)j + (AxBy - AyBx)k
Given:
A = i + j
B = 2i - 3j + k
C = 4j - 3k
(i) Calculate (A x B) x C:
First, find the cross product of A and B:
(A x B) = (1 * 1 - 1 * (-3))i - (1 * 1 - 1 * 2)j + (1 * (-3) - 1 * 2)k
(A x B) = 4i + 1j - 5k
Next, take the cross product of (A x B) and C:
((A x B) x C) = (4 * 0 - (-5) * (-3))i - (4 * 4 - (-5) * 0)j + (4 * (-3) - 0 * (-5))k
((A x B) x C) = -15i - 16j - 12k
So, (A x B) x C = -15i - 16j - 12k.
(ii) Calculate A x (B x C):
First, find the cross product of B and C:
(B x C) = (2 * (-3) - (-3) * 4)i - (2 * (-3) - 4 * 4)j + (2 * 4 - (-3) * (-3))k
(B x C) = -10i - 14j - 23k
Next, take the cross product of A and (B x C):
(A x (B x C)) = (1 * (-23) - 1 * (-14))i - (1 * (-10) - 1 * (-23))j + (1 * (-14) - 1 * (-10))k
(A x (B x C)) = -9i + 13j - 4k
So, A x (B x C) = -9i + 13j - 4k.
I hope this explanation helps! Let me know if you have any further questions.