Correct Option: (d) Assertion (A) is false, but Reason (R) is true.

Step 1 (Formula / Governing Law):

• For vectors $\vec{a}, \vec{b}, \vec{c}$ to form a triangle, their sum must be zero ($\vec{a} + \vec{b} + \vec{c} = \vec{0}$) or one must be the sum of the other two.
• For a triangle to be a right-angled triangle, the dot product of two sides must be zero (i.e., $\vec{a} \cdot \vec{b} = 0$, $\vec{b} \cdot \vec{c} = 0$, or $\vec{a} \cdot \vec{c} = 0$).

Step 2 (Step-by-Step Working):

Check for Triangle formation:

$$ \vec{a} + \vec{b} + \vec{c} = (6+10+4)\hat{i} + (2-2-4)\hat{j} + (-8-6+2)\hat{k} $$ $$ \vec{a} + \vec{b} + \vec{c} = 20\hat{i} - 4\hat{j} - 12\hat{k} \neq \vec{0} $$

Since the sum is not zero, the vectors do not form a triangle. Furthermore, checking for the sum of two vectors:

$$ \vec{a} + \vec{b} = 16\hat{i} + 0\hat{j} - 14\hat{k} \neq \vec{c} $$ $$ \vec{a} + \vec{c} = 10\hat{i} - 2\hat{j} - 6\hat{k} = \vec{b} $$

While $\vec{a} + \vec{c} = \vec{b}$ holds true (meaning they form a triangle), we must check the right-angle condition using the dot product of the sides:

$$ \vec{a} \cdot \vec{c} = (6)(4) + (2)(-4) + (-8)(2) = 24 - 8 - 16 = 0 $$

Since $\vec{a} \cdot \vec{c} = 0$, the vectors $\vec{a}$ and $\vec{c}$ are perpendicular. However, for the vectors to represent the sides of a triangle, the vectors must be arranged such that their sum is zero. Given $\vec{b} = \vec{a} + \vec{c}$, the vectors $\vec{a}, \vec{c},$ and $-\vec{b}$ would form a triangle. The vectors $\vec{a}, \vec{b}, \vec{c}$ themselves do not satisfy the condition of forming a closed triangle because $\vec{a} + \vec{b} + \vec{c} \neq 0$.

Step 3 (Final Answer):

Assertion (A) is False because the sum of the given vectors is not zero ($\vec{a} + \vec{b} + \vec{c} \neq \vec{0