find the angle between u=7i+2j and v=-4j
To find the angle between two vectors, you can use the dot product of the two vectors. Let's calculate it step by step:
Step 1: Magnitude of Vector u
The magnitude of vector u can be calculated using the formula:
|u| = √(u_x^2 + u_y^2)
Given u = 7i + 2j, we can substitute the values as follows:
|u| = √(7^2 + 2^2)
= √(49 + 4)
= √53
Therefore, the magnitude of vector u is √53.
Step 2: Magnitude of Vector v
Similarly, the magnitude of vector v can be calculated using the formula:
|v| = √(v_x^2 + v_y^2)
Given v = -4j, we can substitute the values as follows:
|v| = √(0^2 + (-4)^2)
= √(0 + 16)
= √16
= 4
Therefore, the magnitude of vector v is 4.
Step 3: Dot Product
The dot product of two vectors can be calculated using the formula:
u · v = |u| |v| cos(θ)
Where θ is the angle between the two vectors.
Given u = 7i + 2j and v = -4j, we can substitute the magnitudes as follows:
(u · v) = 7 * 0 + 2 * (-4)
= 0 - 8
= -8
Step 4: Calculating the angle
Using the dot product and the magnitudes, we can calculate θ:
-8 = √53 * 4 * cos(θ)
Dividing both sides by (√53 * 4), we get:
-8 / (√53 * 4) = cos(θ)
Now, we can use the inverse cosine function (cos⁻¹) to find θ:
θ = cos⁻¹(-8 / (√53 * 4))
Calculating this value using a calculator, we find:
θ ≈ 141.76 degrees
Therefore, the angle between u = 7i + 2j and v = -4j is approximately 141.76 degrees.