Let A = (0, 1, 5) ,B = (-sqrt 14,5,1) . Find the angle between A and B.
To find the angle between two vectors, we can use the dot product formula:
A · B = |A| |B| cos(θ)
where A · B is the dot product of vectors A and B, |A| and |B| are the magnitudes of vectors A and B, and θ is the angle between them.
First, let's calculate the magnitudes of vectors A and B:
|A| = √(0^2 + 1^2 + 5^2) = √26
|B| = √((-√14)^2 + 5^2 + 1^2) = √(14 + 26) = √40 = 2√10
Next, let's calculate the dot product of vectors A and B:
A · B = (0*-sqrt14) + (1*5) + (5*1) = -√14 + 5 + 5 = 10 - √14
Now, let's substitute the values into the dot product formula and solve for θ:
10 - √14 = √26 * 2√10 * cos(θ)
Divide both sides by √26 * 2√10:
(10 - √14) / (2√26√10) = cos(θ)
Simplify the expression:
(10 - √14) / (2√(260)) = cos(θ)
Now, use a calculator to evaluate the right-hand side expression:
(10 - √14) / (2√(260)) ≈ 0.205
Next, use the inverse cosine function to find the angle θ:
θ ≈ arccos(0.205) ≈ 1.361 radians or 78.015 degrees
Therefore, the angle between vectors A and B is approximately 1.361 radians or 78.015 degrees.
To find the angle between two vectors A and B, you can use the dot product formula:
A • B = |A| * |B| * cos(theta)
where A • B is the dot product, |A| and |B| are the magnitudes of vector A and vector B respectively, and theta is the angle between the two vectors.
Let's calculate the dot product first:
A • B = (0 * -sqrt(14)) + (1 * 5) + (5 * 1)
= 0 - sqrt(14) + 5
= 5 - sqrt(14)
Next, let's calculate the magnitudes of A and B:
|A| = sqrt(0^2 + 1^2 + 5^2)
= sqrt(0 + 1 + 25)
= sqrt(26)
|B| = sqrt((-sqrt(14))^2 + 5^2 + 1^2)
= sqrt(14 + 25 + 1)
= sqrt(40)
Now, substitute the dot product and magnitudes into the original formula:
5 - sqrt(14) = sqrt(26) * sqrt(40) * cos(theta)
To solve for the angle theta, rearrange the equation:
cos(theta) = (5 - sqrt(14)) / (sqrt(26) * sqrt(40))
Now, calculate the value of cos(theta):
cos(theta) ≈ 0.0937
Finally, find the angle theta by taking the inverse cosine (cos^-1) of cos(theta):
theta ≈ cos^-1(0.0937)
Using a calculator, the approximate value of theta is:
theta ≈ 84.83 degrees.
Therefore, the angle between vectors A and B is approximately 84.83 degrees.