Let x→, y→ and z→ be three vectors each of magnitude 2 and the angle between each pair of them is π3. If a→ is a non - zero vector perpendicular to x→ and y→ ×z→ and b→ is a non - zero vector perpendicular to y→ and z→ ×x→ , then

(A) b→=b→ . z→ z→- x→

(B) a→=a→ . y→ y→- z→

(C) a→ . b→= -a→ . y→ b→ . z→

(D) a→=a→ . y→ z→- y→

JEE Advanced · Mathematics · 30. Vector Algebra

Let $\vec{x}, \vec{y}$ and $\vec{z}$ be three vectors each of magnitude $\sqrt{2}$ and the angle between each pair of them is $\frac{π}{3} .$ If $\vec{a}$ is a non - zero vector perpendicular to $\vec{x}$ and $\vec{y} \times \vec{z}$ and $\vec{b}$ is a non - zero vector perpendicular to $\vec{y}$ and $\vec{z} \times \vec{x}$ , then

A $\vec{b} = (\vec{b} . \vec{z}) (\vec{z} - \vec{x})$
B $\vec{a} = (\vec{a} . \vec{y}) (\vec{y} - \vec{z})$
C $\vec{a} . \vec{b} = - (\vec{a} . \vec{y}) (\vec{b} . \vec{z})$
D $\vec{a} = (\vec{a} . \vec{y}) (\vec{z} - \vec{y})$

Verified Solution

Answer & Solution

Correct Answer

(A) $\vec{b} = (\vec{b} . \vec{z}) (\vec{z} - \vec{x})$

Step-by-step Solution

Detailed explanation

\vec{a}

is in direction of

\vec{x} \times (\vec{y} \times \vec{z})

i.e.,

(\vec{x} . \vec{z}) \vec{y} - (\vec{x} . \vec{y}) \vec{z}

\Rightarrow

\vec{a} = λ_{1} [2 \times \frac{1}{2} (\vec{y} - \vec{z})]

\vec{a} = λ_{1} (\vec{y} - \vec{z})

........(i)
Now

\vec{a} . \vec{y} = λ_{1} (\vec{y} . \vec{y} - \vec{y} . \vec{z})

= λ_{1} (2 - 1) \Rightarrow λ_{1} = \vec{a} . \vec{y}

........(ii)
From (i) and (ii),

\vec{a} = \vec{a} . \vec{y} (\vec{y} - \vec{z})

Similarly,

\vec{b} = (\vec{b} . \vec{z}) (\vec{z} - \vec{x})

Now,

\vec{a} . \vec{b} = (\vec{a} . \vec{y}) (\vec{b} . \vec{z}) [(\vec{y} - \vec{z}) . (\vec{z} - \vec{x})]

= (\vec{a} . \vec{y}) (\vec{b} . \vec{z}) [1 - 1 - 2 + 1]

= - (\vec{a} . \vec{y}) (\vec{b} . \vec{z})

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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