Find the angle between the vectors i + j and i + 2j-3k.
Time to review the dot product.
u•v = |u| * |v| * cosθ
So, just solve
√2 * √14 cosθ = 1*1 + 1*2 - 0*3
To find the angle between two vectors, you can use the dot product formula and the magnitude of the vectors. Here are the steps to find the angle between the vectors i + j and i + 2j - 3k:
Step 1: Calculate the dot product of the two vectors.
The dot product of two vectors A and B is defined as:
A · B = |A| |B| cos(theta)
where |A| and |B| are the magnitudes of vectors A and B, and theta is the angle between them.
For the vectors i + j and i + 2j - 3k, the dot product can be calculated as follows:
(i + j) · (i + 2j - 3k) = (1 * 1) + (1 * 2) + (0 * -3) = 1 + 2 + 0 = 3
Step 2: Calculate the magnitudes of the vectors.
The magnitude of a vector A is given by the formula:
|A| = sqrt(A · A)
where A · A represents the dot product of vector A with itself.
For the vectors i + j and i + 2j - 3k, the magnitudes can be calculated as follows:
| i + j | = sqrt((1 * 1) + (1 * 1)) = sqrt(2)
| i + 2j - 3k | = sqrt((1 * 1) + (2 * 2) + (-3 * -3)) = sqrt(1 + 4 + 9) = sqrt(14)
Step 3: Calculate the cosine of the angle.
Using the dot product and magnitudes calculated in the previous steps, substitute them into the dot product formula:
3 = (sqrt(2))(sqrt(14)) cos(theta)
Step 4: Solve for theta.
To find the value of theta, divide both sides of the equation by (sqrt(2))(sqrt(14)):
cos(theta) = 3 / (sqrt(2))(sqrt(14))
Now, take the inverse cosine (arccos) of both sides to find the angle:
theta = arccos(3 / (sqrt(2))(sqrt(14)))
Calculating the value using a calculator or software, you will find:
theta ≈ 39.23 degrees (rounded to two decimal places)
Therefore, the angle between the vectors i + j and i + 2j - 3k is approximately 39.23 degrees.