Calculation of linear velocity |v|

To determine the linear velocity |v| of the rotating body, we will use the formula v = w × r, where v is the linear velocity, w is the angular velocity, and r is the radius vector.

Given data:
Angular velocity, w = 5i - 4j + 9k
Radius vector, r = 8i - 6j + 3k

Step 1: Calculate the cross product of w and r
The cross product of two vectors w and r is given by:
w × r = (w2*r3 - w3*r2)i + (w3*r1 - w1*r3)j + (w1*r2 - w2*r1)k

Substituting the given values:
w × r = ((-4)*(3) - (9)*(-6))i + ((9)*(8) - (5)*(3))j + ((5)*(-6) - (-4)*(8))k
= (-12 + 54)i + (72 - 15)j + (-30 + 32)k
= 42i + 57j + 2k

Step 2: Calculate the magnitude of the cross product |w × r|
The magnitude of a vector is given by:
|v| = √(v1^2 + v2^2 + v3^2)

Substituting the calculated values:
|w × r| = √((42)^2 + (57)^2 + (2)^2)
= √(1764 + 3249 + 4)
= √6017
≈ 77.65

Therefore, the magnitude of the linear velocity |v| is approximately 77.65.

Explanation:
The linear velocity of a rotating body is determined by the cross product of the angular velocity and the radius vector. The cross product results in a vector perpendicular to both the angular velocity and radius vector. The magnitude of this vector represents the magnitude of the linear velocity |v|.

In this case, we calculated the cross product of w = 5i - 4j + 9k and r = 8i - 6j + 3k using the formula w × r = (w2*r3 - w3*r2)i + (w3*r1 - w1*r3)j + (w1*r2 - w2*r1). After substituting the given values, we obtained the cross product vector as 42i + 57j + 2k.

To find the magnitude of the cross product, we used the formula |v| = √(v1^2 + v2^2 + v3^2), where v1, v2, and v3 represent the components of the cross product vector. After substituting the calculated values, we found the magnitude of the linear velocity |v| to be approximately 77.65.

Take cross of w and r and find mod of v...
Try urself first....