Let a be a unit vector perpendicular to the vectors b=i-2j+3k and c=2i+3j-k, and makes an angle of cos ⁻¹(-1/3) with the vector i+j+k. If a makes an angle of (π)/3 with the vector i+ +k, then the value of α is:

Vector Algebra: JEE Main 2025 Mathematics question

JEE Main 2025 · January, Unspecified (session-level dataset) · Q127

JEE Main 2025January, Unspecified (session-level dataset)MathematicsMediumVector Algebra184× since 2002

Let

\hat{a}

be a unit vector perpendicular to the vectors

\vec{b}=\hat{i}-2\hat{j}+3\hat{k}

and

\vec{c}=2\hat{i}+3\hat{j}-\hat{k}

, and makes an angle of

\cos^{-1}\left(-\frac{1}{3}\right)

with the vector

\hat{i}+\hat{j}+\hat{k}

. If

\hat{a}

makes an angle of

\frac{\pi}{3}

with the vector

\hat{i}+\alpha\hat{j}+\hat{k}

, then the value of

\alpha

is:

JEE Main 2025 · January, Unspecified (session-level dataset) · Q127

JEE Main 2025January, Unspecified (session-level dataset)MathematicsMediumVector Algebra184× since 2002

Let

\hat{a}

be a unit vector perpendicular to the vectors

\vec{b}=\hat{i}-2\hat{j}+3\hat{k}

and

\vec{c}=2\hat{i}+3\hat{j}-\hat{k}

, and makes an angle of

\cos^{-1}\left(-\frac{1}{3}\right)

with the vector

\hat{i}+\hat{j}+\hat{k}

. If

\hat{a}

makes an angle of

\frac{\pi}{3}

with the vector

\hat{i}+\alpha\hat{j}+\hat{k}

, then the value of

\alpha

is: