Find the angles between the following pairs of vectors

(1)3i+2j-6k,4i-3j+k
(2)2i-3j+k,3i-j-2k

To find the angles between the given pairs of vectors, we can use the dot product formula and the magnitude formula. Here are the step-by-step explanations for each pair of vectors:

(1) 3i + 2j - 6k and 4i - 3j + k:

Step 1: Calculate the dot product of the vectors.
The dot product is calculated by multiplying the corresponding components of the vectors and then adding them together. In this case, the dot product is given by:
(3 * 4) + (2 * -3) + (-6 * 1) = 12 - 6 - 6 = 0

Step 2: Calculate the magnitudes of the vectors.
The magnitude of a vector is calculated by taking the square root of the sum of the squares of its components. In this case, the magnitudes are:
||3i + 2j - 6k|| = sqrt((3^2) + (2^2) + (-6^2)) = sqrt(9 + 4 + 36) = sqrt(49) = 7
||4i - 3j + k|| = sqrt((4^2) + (-3^2) + (1^2)) = sqrt(16 + 9 + 1) = sqrt(26)

Step 3: Calculate the angle between the vectors.
The angle between the vectors can be found using the formula:
cos(theta) = (dot product of vectors) / (product of magnitudes of vectors)
Using the values calculated in Step 1 and Step 2, we have:
cos(theta) = 0 / (7 * sqrt(26))

Step 4: Calculate the angle.
To find the angle between the vectors, we can take the arccosine of the value calculated in Step 3. Using a calculator, we can find:
theta ≈ 90 degrees

So, the angle between the vectors (3i + 2j - 6k) and (4i - 3j + k) is approximately 90 degrees.

(2) 2i - 3j + k and 3i - j - 2k:

Following the same steps as above, we can determine the angle between the vectors (2i - 3j + k) and (3i - j - 2k).

Step 1: Calculate the dot product of the vectors.
(2 * 3) + (-3 * -1) + (1 * -2) = 6 + 3 - 2 = 7

Step 2: Calculate the magnitudes of the vectors.
||2i - 3j + k|| = sqrt((2^2) + (-3^2) + (1^2)) = sqrt(4 + 9 + 1) = sqrt(14)
||3i - j - 2k|| = sqrt((3^2) + (-1^2) + (-2^2)) = sqrt(9 + 1 + 4) = sqrt(14)

Step 3: Calculate the angle between the vectors.
cos(theta) = (dot product of vectors) / (product of magnitudes of vectors)
cos(theta) = 7 / (sqrt(14) * sqrt(14)) = 7 / 14 = 1/2

Step 4: Calculate the angle.
theta = arccos(1/2) ≈ 60 degrees

So, the angle between the vectors (2i - 3j + k) and (3i - j - 2k) is approximately 60 degrees.