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When we talk about velocity vectors, we are discussing both the speed and direction of a moving object. A velocity vector contains information about how fast an object is moving along each axis in a 3D space. For instance, the velocity vector \(3 \hat{i}-2 \hat{j}\) signifies a motion where there is movement of 3 units in the direction of \(\hat{i}\), and a motion of -2 units along \(\hat{j}\).
To understand velocity, it's important to grasp the concept of direction as well. Directions are represented by unit vectors, such as \(\hat{i}, \hat{j},\) and \(\hat{k}\), which correspond to the x, y, and z axes respectively.
Velocity vectors provide a visual and mathematical way to analyze and calculate an object’s motion, making them vital for problems in physics, particularly involving collisions.

Collision mechanics involves the study of interactions between two or more bodies that come into contact and exert forces on each other. In this context, particles collide and stick together, which is known as a perfectly inelastic collision. In the given problem, the two particles have a glued impact, which means they move together after colliding.
Important concepts to consider are:

• Conservation of Momentum: The total momentum before the collision is equal to the total momentum after the collision. This principle acts as a reliable tool to predict the aftermath of a collision.
• Relative Speed: After a perfectly inelastic collision, the relative speed of the two colliding objects is zero since they stick together.

The final outcome, a single particle, can be imagined as a union of the two original particles exerting combined forces, characterized by a cumulative velocity.

Momentum calculations are pivotal for determining the velocity after a collision. The momentum of an object is a product of its mass and velocity, expressed as \(\vec{p} = m \vec{v}\). Total momentum before and after the collision should remain constant due to the conservation of momentum law.
In the proposed problem, calculate the total momentum before collision using the equation \(m_1 \vec{v_1} + m_2 \vec{v_2} = (m_1 + m_2) \vec{V}\).

After plugging in given values:

• \(m_1 = 1\) g and \(\vec{v_1} = 3 \hat{i}-2 \hat{j}\)
• \(m_2 = 2\) g and \(\vec{v_2} = 4 \hat{j}-6 \hat{k}\)

Sum these to find the total initial momentum, and solve for \(\vec{V}\) using basic algebra.
Eventually, find the resultant speed by computing the magnitude of the velocity vector \(\vec{V}\) using \(\| \vec{V} \| = \sqrt{\vec{V}\cdot\vec{V}}\). This will yield the final speed of the combined particle.