If D=5i+6j-3k and E =7i +8j +4k
Calculate,DxE
Find the angle between D and E
Show the DxEis perpendicular to both D and E
I need help please
D•E = |D| * |E|* cosθ
Just evaluate the determinant as usual.
To calculate DxE, we need to find the cross product of the vectors D and E.
The cross product of two vectors, A and B, can be calculated using the formula:
AxB = (AyBz - AzBy)i - (AxBz - AzBx)j + (AxBy - AyBx)k
Let's substitute the values of D and E into the formula:
DxE = ((6 * 4) - (-3 * 8))i - ((5 * 4) - (-3 * 7))j + ((5 * 8) - (6 * 7))k
Simplifying this expression:
DxE = (24 + 24)i - (20 - 21)j + (40 - 42)k
Finally, we have:
DxE = 48i + 1j - 2k
Now, let's find the angle between D and E using the dot product formula:
The dot product of two vectors, A and B, can be calculated using the formula:
A · B = |A| |B| cosθ
Let's substitute the values of D and E into the formula:
D · E = (5 * 7) + (6 * 8) + (-3 * 4)
Simplifying this expression:
D · E = 35 + 48 - 12
D · E = 71
The magnitudes of D and E can be calculated as:
|D| = √(5^2 + 6^2 + (-3)^2) = √(25 + 36 + 9) = √70
|E| = √(7^2 + 8^2 + 4^2) = √(49 + 64 + 16) = √129
Now, we can calculate the cosine of the angle:
cosθ = (D · E) / (|D||E|) = (71) / (√70 * √129)
Therefore, the angle between D and E can be found using the inverse cosine function:
θ = cos^-1[(71) / (√70 * √129)]
To determine if DxE is perpendicular to both D and E, we can take the dot product between DxE and D, as well as DxE and E. If both dot products are zero, then DxE will be perpendicular to both D and E.
Let's calculate the dot product between DxE and D:
(DxE) · D = (48 * 5) + (1 * 6) + (-2 * -3)
Simplifying this expression:
(DxE) · D = 240 + 6 + 6
(DxE) · D = 252
Since the dot product is not zero, DxE is not perpendicular to D.
Now, let's calculate the dot product between DxE and E:
(DxE) · E = (48 * 7) + (1 * 8) + (-2 * 4)
Simplifying this expression:
(DxE) · E = 336 + 8 - 8
(DxE) · E = 336
Similarly, the dot product is not zero, so DxE is not perpendicular to E either.
Therefore, DxE is not perpendicular to both D and E.
To calculate the cross product DxE, we can use the cross product formula:
D x E = (DyEz - DzEy)i - (DxEz - DzEx)j + (DxEy - DyEx)k
Let's substitute the values of D and E into the formula:
DxE = ((6 * 4) - (-3 * 8))i - ((5 * 4) - (-3 * 7))j + ((5 * 8) - (6 * 7))k
= (24 + 24)i - (20 - 21)j + (40 - 42)k
= 48i - (-1)j - 2k
= 48i + j - 2k
So, D x E = 48i + j - 2k.
To find the angle between D and E, we can use the dot product formula:
|D·E| = |D| * |E| * cosθ
Where |D·E| is the magnitude of the dot product, |D| and |E| are the magnitudes of vectors D and E, and θ is the angle between them.
Let's substitute the given values into the formula:
|D·E| = |D| * |E| * cosθ
= √(Dx^2 + Dy^2 + Dz^2) * √(Ex^2 + Ey^2 + Ez^2) * cosθ
= √((5^2) + (6^2) + (-3^2)) * √((7^2) + (8^2) + (4^2)) * cosθ
= √(25 + 36 + 9) * √(49 + 64 + 16) * cosθ
= √(70) * √(129) * cosθ
= √(70 * 129) * cosθ
= √(9030) * cosθ
Now, let's calculate the magnitude of D·E:
D·E = DxE_x + DxE_y + DxE_z
= (5 * 7) + (6 * 8) + (-3 * 4)
= 35 + 48 - 12
= 71
Substituting this back into the equation:
71 = √(9030) * cosθ
To find cosθ, we can divide both sides by √(9030):
cosθ = 71 / √(9030)
Using a calculator, we can find the value of cosθ:
cosθ ≈ 0.933
To find the angle θ, we can use the inverse cosine function (cos^(-1)):
θ ≈ cos^(-1)(0.933)
θ ≈ 22.91 degrees
Therefore, the angle between D and E is approximately 22.91 degrees.
To show that D x E is perpendicular to both D and E, we can calculate the dot product of D x E with D and E separately. If the dot product is zero, it indicates that the two vectors are perpendicular.
Let's calculate:
D x E · D = (48 * 5) + (1 * 6) + (-2 * (-3))
= 240 + 6 + 6
= 252
D x E · E = (48 * 7) + (1 * 8) + (-2 * 4)
= 336 + 8 - 8
= 336
Since D x E · D ≠ 0 and D x E · E ≠ 0, we can conclude that the D x E vector is not perpendicular to both D and E.