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 magnitude of the resultant and the angle with the direction of Q
Let's put q on the x axis
then p is 30 degrees above the x axis
In the direction of q (x direction):
3 + 5 cos 30 = 3 + 4.33 = 7.33
in the direction perpendicular to q (y axis) :
5 sin 30 = 2.5
|R| = sqrt (7.33^2 + 2.5^2) = sqrt(60) = 7.74
tan angle above x axis = 2.5 / 7.33 = 0.341
angle = 18.8 degrees
To calculate the resultant vector, we can use the vector addition formula:
Resultant = √(p^2 + q^2 + 2pqcosθ)
where p and q are the magnitudes of vectors p and q, and θ is the angle between them.
i) To determine the resultant vector in the direction of q:
Substituting the given values into the formula,
Resultant = √(5^2 + 3^2 + 2(5)(3)cos30°)
= √(25 + 9 + 30)
= √64
= 8
Therefore, the magnitude of the resultant vector in the direction of q is 8 units.
ii) To calculate the resultant vector perpendicular to the direction of q:
We can use the formula:
Resultant = 2pqsinθ
Substituting the given values,
Resultant = 2(5)(3)sin30°
= 30(sin30°)
= 30(0.5)
= 15
Therefore, the magnitude of the resultant vector perpendicular to the direction of q is 15 units.
b) To calculate the magnitude and direction of the resultant:
We can use the Pythagorean theorem to find the magnitude of the resultant:
Magnitude of the resultant = √[(Resultant in the direction of q)^2 + (Resultant perpendicular to the direction of q)^2]
= √[(8^2) + (15^2)]
= √(64 + 225)
= √289
= 17
Therefore, the magnitude of the resultant vector is 17 units.
To find the angle with the direction of q, we can use the inverse tangent function:
Angle = tan^(-1)[(Resultant perpendicular to the direction of q)/(Resultant in the direction of q)]
= tan^(-1)(15/8)
≈ 60.94°
Therefore, the angle with the direction of q is approximately 60.94°.
To calculate the resultant vector in the direction of vector q, we need to find the vector component of vector p in the direction of q. This can be done using the formula:
Resultant in the direction of q = magnitude of vector p * cos(angle between p and q) * unit vector in the direction of q
Let's use this formula to find the resultant in the direction of q:
i) Resultant in the direction of q:
Magnitude of vector p = 5 units
Magnitude of vector q = 3 units
Angle between p and q = 30 degrees
Resultant in the direction of q = 5 * cos(30) * unit vector in the direction of q
To calculate the unit vector in the direction of vector q, we divide vector q by its magnitude:
Unit vector in the direction of q = vector q / magnitude of vector q
Unit vector in the direction of q = q / 3
Substituting the values:
Resultant in the direction of q = 5 * cos(30) * (q / 3)
Simplifying the equation, we get:
Resultant in the direction of q = (5/2) * (q / 3)
ii) To calculate the resultant perpendicular to the direction of q, we need to find the vector component of vector p perpendicular to vector q. This can be done using the formula:
Resultant perpendicular to the direction of q = magnitude of vector p * sin(angle between p and q) * unit vector perpendicular to q
Let's use this formula to find the resultant perpendicular to the direction of q:
Resultant perpendicular to the direction of q = 5 * sin(30) * unit vector perpendicular to q
To calculate the unit vector perpendicular to the direction of vector q, we can take the negative reciprocal of the gradient of q:
Unit vector perpendicular to q = -1 / (gradient of q)
The gradient of q is given by:
gradient of q = qy / qx
Where qy and qx are the y and x components of vector q.
Substituting the values, we get:
gradient of q = 0 / 3 = 0
Therefore, the unit vector perpendicular to q is undefined.
b) The magnitude of the resultant vector can be calculated using the Pythagorean theorem:
Magnitude of the resultant = sqrt(Resultant in the direction of q^2 + Resultant perpendicular to the direction of q^2)
Substituting the values, we get:
Magnitude of the resultant = sqrt((5/2) * (q / 3)^2 + 0^2)
Simplifying the equation, we get:
Magnitude of the resultant = sqrt((25/4) * (q^2 / 9))
Finally, the angle with the direction of vector q can be calculated using the formula:
Angle with the direction of q = arctan(Resultant perpendicular to the direction of q / Resultant in the direction of q)
Substituting the values, we get:
Angle with the direction of q = arctan(0 / (5/2) * (q / 3))
Simplifying the equation, we get:
Angle with the direction of q = arctan(0)
Therefore, the angle with the direction of q is 0 degrees.