Animated Solution for Mathematics - Vector Algebra: The non-zero vectors are $\vec{a},\vec{b}$ and $\vec{c}$ are related by $\vec{a}=8\vec{b}$ and $\vec{c}=-7\vec{b}$. Then the angle between $\vec{a}$ and $\vec{c}$ is

Welcome, fellow traveler on the path to JEE excellence. Today, we are not just solving a problem; we are witnessing the elegant dance of vectors in space.

Often, when we see equations like $\vec{a}=8\vec{b}$ and $\vec{c}=-7\vec{b}$, our minds immediately jump to complex calculations. But stop for a moment. Breathe.

Let us look at the geometry, the soul of the problem. Imagine you are standing at the origin of a coordinate system with a single, fundamental arrow pointing somewhere in space, which we call $\vec{b}$. This is your anchor, your reference point. Everything in this problem is defined by its relationship to this single, humble vector.

Now, consider the magic of scalar multiplication. When you multiply a vector by a number, you are essentially performing a transformation.

If you multiply $\vec{b}$ by a positive number like $8$, you are simply stretching it. The direction stays exactly the same. It is like looking at a photograph and zooming in; the subject is larger, but it is still pointing in the same direction.

Thus, $\vec{a}=8\vec{b}$ tells us that $\vec{a}$ is just a longer version of $\vec{b}$, marching in perfect harmony with it.

But what happens when that number is negative? This is the trap that catches many students. When you multiply by $-7$, the negative sign acts like a mirror.

It does not just scale the length; it flips the direction entirely. It is a perfect $180^{\circ }$ rotation. So, $\vec{c}=-7\vec{b}$ is not just a scaled version of $\vec{b}$; it is an anti-parallel version, pointing in the exact opposite direction.

Let us visualize this. Imagine $\vec{b}$ is pointing to the right. Because $\vec{a}=8\vec{b}$, $\vec{a}$ is also pointing to the right.

Now, look at $\vec{c}=-7\vec{b}$. Because of that negative sign, $\vec{c}$ is pointing to the left.

If you stand at the origin and look at $\vec{a}$ and $\vec{c}$, you see them pointing in diametrically opposite directions. They are like two people standing back-to-back, walking away from each other.

In the language of geometry, when two vectors point in exactly opposite directions, the angle between them is a straight line. A straight line, as we know from our earliest geometry lessons, corresponds to an angle of $180^{\circ }$.

In the world of advanced physics and mathematics, we prefer to measure this rotation in radians. We know that a full circle is $2\pi$ radians, which is $360^{\circ }$.

Therefore, a half-circle, or a straight angle, is exactly $\pi$ radians.

We did not need complex dot products or coordinate geometry to find the answer. We simply needed to understand the physical meaning of the scalar multipliers.

The angle between $\vec{a}$ and $\vec{c}$ is $\pi$.

Keep this intuition with you. Whenever you see a vector equation, do not just calculate; visualize. See the vectors as arrows, see the scalars as transformations, and the geometry will reveal itself to you. You are doing great—keep pushing, keep visualizing, and the physics will become second nature.