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Question: Given, \(\vec \omega = 2\hat k\)and \(\hat r = 2\hat i + 2\hat j\). Find the linear velocity....

Given, ω⃗=2k^\vec \omega = 2\hat kand r^=2i^+2j^\hat r = 2\hat i + 2\hat j. Find the linear velocity.

Explanation

Solution

There are majorly two types of quantities, scalar and vector quantities. Also, we have two methods of finding the product of two vectors; Scalar or dot product and vector or cross product. Scalar quantities are the result of dot product whereas vector quantities are the result of cross product. Since linear velocity is a vector quantity, we have to express it by the cross product of two different vectors.

Complete step-by-step answer:
The vector product of two vectors is also called a cross product. It’s called a vector product as its result is a vector quantity. For example the vector product of force and distance from the axis of rotation gives a vector quantity called torque. τ=F⃗×r⃗\tau = \vec F \times \vec r. Similarly, the relation between angular speed and linear velocity is v⃗=ω⃗×r⃗\vec v = \vec \omega \times \vec r.
Vector product of two vectors A⃗ and B⃗ is A⃗×B⃗= ∣A∣∣B∣sinθ\vec A \ and\ \vec B \ is \ \vec A\times \vec B =\ |A||B|sin\theta, where θ\theta is the angle between the two vectors. This form could be used if the angles between two vectors is known.
But generally we can directly calculate the cross product if the quantities ‘A’ and ‘B’ are given in the form of vectors.
Hence v⃗=ω⃗×r⃗\vec v = \vec \omega \times \vec r
  ⟹  v⃗=(2k^)×(2i^+2j^)\implies \vec v = (2\hat k) \times (2\hat i + 2\hat j)
  ⟹  v⃗=4j^−4i^\implies \vec v = 4\hat j - 4\hat i [as k^×i^=k^×j^=−i^\hat k \times \hat i = \hat k \times \hat j = -\hat i]
Hence, the linear velocity vector is v⃗=4j^−4i^\vec v = 4\hat j - 4\hat i and hence the magnitude of the linear velocity is 42+42=42 m/s\sqrt{4^2+4^2} = 4\sqrt2 \ m/s

Additional Information: While representing a vector quantity, we have to represent both its direction, (using unit vector) and the magnitude. This is the standard way of representing vectors.

Note: There are certain vector quantities which are resulted by product of other vector quantities. Like we can’t add vectors by traditional addition (scalar addition), similarly they can’t be multiplied either. Hence we have two types of methods to find the product of vectors, scalar and vector. Scalar product is used if the resultant quantity is a scalar quantity and vector product is used if the resultant quantity is a vector quantity.