Two vectors p and q of magnitude 5 and 3 units respectively are inclined at an angle of 30o to each other.calculate the resultant
i) in the direction of q
ii) perpendicular to the direction of q
b) hence calculate the magnitude of the resultant and the angle it makes with the direction of q.
To calculate the resultant, we can use vector addition.
First, let's represent vector p as P and vector q as Q. We are given that the magnitude of vector P is 5 units, and the magnitude of vector Q is 3 units. Additionally, we are told that the angle between the two vectors is 30 degrees.
i) To calculate the resultant in the direction of Q, we need to find the component of vector P in the direction of Q. This can be done using the formula:
PQ = P * cos(angle)
where angle is the angle between the two vectors.
Using the given values:
PQ = 5 * cos(30)
Now, calculate the value of PQ:
PQ = 5 * 0.866
PQ ≈ 4.33 units
Therefore, the resultant in the direction of Q is approximately 4.33 units.
ii) To calculate the resultant perpendicular to the direction of Q, we need to find the component of vector P perpendicular to Q. This can be done using the formula:
P⊥ = P * sin(angle)
Again, using the given values:
P⊥ = 5 * sin(30)
Now, calculate the value of P⊥:
P⊥ = 5 * 0.5
P⊥ = 2.5 units
Therefore, the resultant perpendicular to the direction of Q is 2.5 units.
b) To calculate the magnitude of the resultant and the angle it makes with the direction of Q, we can use the Pythagorean theorem and trigonometry.
The magnitude of the resultant can be found using the formula:
Resultant = sqrt[(PQ)^2 + (P⊥)^2]
Using the calculated values:
Resultant = sqrt[(4.33)^2 + (2.5)^2]
Resultant ≈ sqrt[18.7489 + 6.25]
Resultant ≈ sqrt[24.9989]
Resultant ≈ 4.99 units
The angle the resultant makes with the direction of Q can be found using the formula:
θ = atan(P⊥ / PQ)
Using the calculated values:
θ = atan(2.5 / 4.33)
θ ≈ atan(0.577)
Therefore, the magnitude of the resultant is approximately 4.99 units, and the angle it makes with the direction of Q is approximately 30.96 degrees.