Question

Determine the angle that vector a= (12,-3,4) makes with the y-axis.

Answer 1

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|)

Answer 2

[AI-generated answer — Explain Bot]

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: 𝑎 ⋅ 𝑏 = |𝑎| |𝑏| cos(θ), where 𝑎 ⋅ 𝑏 represents the dot product of vectors 𝑎 and 𝑏, |𝑎| represents the magnitude (length) of vector 𝑎, and θ 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 𝑏 = (0, 1, 0), which has a magnitude of 1.

So, 𝑎 ⋅ 𝑏 = (12,-3,4) ⋅ (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: |𝑎| = √(𝑎₁² + 𝑎₂² + 𝑎₃²), where 𝑎₁, 𝑎₂, 𝑎₃ represent the components of vector a.

For vector a = (12,-3,4), |𝑎| = √(12² + (-3)² + 4²) = √(144 + 9 + 16) = √(169) = 13.

Step 3: Calculate the angle θ
Using the formula 𝑎 ⋅ 𝑏 = |𝑎| |𝑏| cos(θ), we can rearrange it to solve for θ: θ = cos⁻¹(𝑎 ⋅ 𝑏 / (|𝑎| |𝑏|)).

In this case, θ = cos⁻¹(-3 / (13 * 1)) = cos⁻¹(-3 / 13) ≈ 105.28 degrees.

Therefore, the angle that vector a makes with the y-axis is approximately 105.28 degrees.