Determine the angle that vector a= (12,-3,4) makes with the y-axis.
A vector B along y axis could be 0i + 1 j + 0 k
Then what is A dot B ?
B = 0 i + 1 j + 0 k
A = 12i - 3 j + 4 k
A dot B = 0 -3 + 0 = -3
|A| = sqrt (144+9+16)
|B| = 1
cos (T) = -3/(|A||B|)
To determine the angle that vector a makes with the y-axis, we can use the dot product between vector a and the unit vector in the y-direction.
Step 1: Calculate the dot product
The dot product between two vectors is given by the formula: π β
π = |π| |π| cos(ΞΈ), where π β
π represents the dot product of vectors π and π, |π| represents the magnitude (length) of vector π, and ΞΈ represents the angle between the vectors.
In this case, we want to find the dot product of vector a and the unit vector in the y-direction. The unit vector in the y-direction is π = (0, 1, 0), which has a magnitude of 1.
So, π β
π = (12,-3,4) β
(0, 1, 0) = 0 + (-3) + 0 = -3.
Step 2: Calculate the length of vector a
The magnitude (length) of vector a is given by the formula: |π| = β(πβΒ² + πβΒ² + πβΒ²), where πβ, πβ, πβ represent the components of vector a.
For vector a = (12,-3,4), |π| = β(12Β² + (-3)Β² + 4Β²) = β(144 + 9 + 16) = β(169) = 13.
Step 3: Calculate the angle ΞΈ
Using the formula π β
π = |π| |π| cos(ΞΈ), we can rearrange it to solve for ΞΈ: ΞΈ = cosβ»ΒΉ(π β
π / (|π| |π|)).
In this case, ΞΈ = cosβ»ΒΉ(-3 / (13 * 1)) = cosβ»ΒΉ(-3 / 13) β 105.28 degrees.
Therefore, the angle that vector a makes with the y-axis is approximately 105.28 degrees.