A line makes angles α, β and γ with the positive directions of the coordinate axes. If $\backslash (\backslash vec\{a\}=(\backslash sin^2\; \backslash alpha)\backslash hat\{i\}\; +\; (\backslash sin^2\; \backslash beta)\backslash hat\{j\}\; +\; (\backslash sin^2\; \backslash gamma)\backslash hat\{k\}\; \backslash text\{\; and\; \}\; \backslash vec\{b\}\; =\; \backslash hat\{i\}\; +\; \backslash hat\{j\}\; +\; \backslash hat\{k\}\backslash )$ , then what is \(\vec{a}.\vec{b}\) equal to?

Calculation:

Given,

\( \cos^2(\alpha) + \cos^2(\beta) + \cos^2(\gamma) = 1 \)

Using the identity \( \cos^2(x) = 1 - \sin^2(x) \), we substitute:

\( (1 - \sin^2(\alpha)) + (1 - \sin^2(\beta)) + (1 - \sin^2(\gamma)) = 1 \)

Simplifying the equation:

\( 3 - (\sin^2(\alpha) + \sin^2(\beta) + \sin^2(\gamma)) = 1 \)

Rearrange to isolate the sine terms:

\( \sin^2(\alpha) + \sin^2(\beta) + \sin^2(\gamma) = 2 \)

Now, calculate the dot product:

\( \vec{a} \cdot \vec{b} = \sin^2(\alpha) + \sin^2(\beta) + \sin^2(\gamma) = 2 \)

∴ The value of \( \vec{a} \cdot \vec{b} \)is 2.

Hence, the correct answer is Option 4.