For the following 3 vectors
A⃗ =2y^+3z^
B⃗ = 3 x^+2z^
C⃗ = 3 x^+3y^
Calculate the following:
(a) A⃗ ⋅(B⃗ +C⃗ )=
(b) D⃗ =A⃗ ×(B⃗ +C⃗ )
Dx=
Dy=
Dz=
(c) A⃗ ⋅(B⃗ ×C⃗ )=
(d) G⃗ =A⃗ ×(B⃗ ×C⃗ )
Gx=
Gy=
Gz=
To calculate the given vectors and their dot products or cross products, we can follow these steps:
(a) A⃗ ⋅ (B⃗ + C⃗ ):
- Start by finding (B⃗ + C⃗ ) by adding the corresponding components together:
(B⃗ + C⃗ ) = (3x^ + 2z^) + (3x^ + 3y^)
= 3x^ + 2z^ + 3x^ + 3y^
= (3x^ + 3x^) + (3y^) + 2z^
= 6x^ + 3y^ + 2z^
- Now we can calculate the dot product of A⃗ and (B⃗ + C⃗ ):
A⃗ ⋅ (B⃗ + C⃗ ) = (2y^ + 3z^) ⋅ (6x^ + 3y^ + 2z^)
To calculate the dot product, we multiply the corresponding components and sum them:
A⃗ ⋅ (B⃗ + C⃗ ) = (2 * 0) + (3 * 3) + (0 * 2)
= 9
Therefore, A⃗ ⋅ (B⃗ + C⃗ ) = 9.
(b) D⃗ = A⃗ × (B⃗ + C⃗ ):
- Start by finding (B⃗ + C⃗ ) by adding the corresponding components together:
(B⃗ + C⃗ ) = (3x^ + 2z^) + (3x^ + 3y^)
= 3x^ + 2z^ + 3x^ + 3y^
= (3x^ + 3x^) + (3y^) + 2z^
= 6x^ + 3y^ + 2z^
Now we can calculate the cross product of A⃗ and (B⃗ + C⃗ ):
D⃗ = A⃗ × (B⃗ + C⃗ ) = (2y^ + 3z^) × (6x^ + 3y^ + 2z^)
To calculate the cross product, we can use the determinant method:
Dx = (3z^ * 3y^) - (3y^ * 2z^) = (9z^y^) - (6z^y^) = 3z^y^
Dy = (2z^ * 6x^) - (3z^ * 3x^) = (12z^x^) - (9z^x^) = 3z^x^
Dz = (2y^ * 3x^) - (3z^ * 6x^) = (6yx^) - (18zx^) = -12zx^ + 6yx^
Therefore, Dx = 3z^y^, Dy = 3z^x^, and Dz = -12zx^ + 6yx^.
(c) A⃗ ⋅ (B⃗ × C⃗ ):
Now we need to calculate the cross product of B⃗ and C⃗ :
B⃗ × C⃗ = (3x^ + 2z^) × (3x^ + 3y^)
Again, using the determinant method:
(B⃗ × C⃗ ) = (2z^ * 3y^) - (3x^ * 3y^) = (6z^y^) - (9x^y^) = -9x^y^ + 6z^y^
Now we can calculate the dot product of A⃗ and (B⃗ × C⃗ ):
A⃗ ⋅ (B⃗ × C⃗ ) = (2y^ + 3z^) ⋅ (-9x^y^ + 6z^y^)
Using the dot product calculation:
A⃗ ⋅ (B⃗ × C⃗ ) = (2 * -9) + (3 * 6) + (0 * 0)
= -18 + 18 + 0
= 0
Therefore, A⃗ ⋅ (B⃗ × C⃗ ) = 0.
(d) G⃗ = A⃗ × (B⃗ × C⃗ ):
First, we need to calculate the cross product of B⃗ and C⃗ :
B⃗ × C⃗ = (3x^ + 2z^) × (3x^ + 3y^)
Using the determinant method:
(B⃗ × C⃗ ) = (2z^ * 3y^) - (3x^ * 3y^) = (6z^y^) - (9x^y^) = -9x^y^ + 6z^y^
Now we can calculate the cross product of A⃗ and (B⃗ × C⃗ ):
G⃗ = A⃗ × (B⃗ × C⃗ ) = (2y^ + 3z^) × (-9x^y^ + 6z^y^)
Using the cross product calculation:
Gx = (3z^ * 6z^y^) - (6z^ * -9x^y^) = (18z^2y^) + (54zx^y^) = 54zx^y^ + 18z^2y^
Gy = (2z^ * -9x^y^) - (3z^ * -9x^y^) = (-18zx^y^) + (27zx^y^) = 9zx^y^
Gz = (2y^ * -9x^y^) - (3z^ * 6z^y^) = (-18x^y^2) - (18zx^y^) = -18x^y^2 - 18zx^y^
Therefore, Gx = 54zx^y^ + 18z^2y^, Gy = 9zx^y^, and Gz = -18x^y^2 - 18zx^y^.