Let O be the origin and $\overrightarrow{\mathrm{OA}}=2 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}}, \overrightarrow{\mathrm{OB}}=\hat{\mathrm{i}}-2 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{OC}}=\frac{1}{2}(\overrightarrow{\mathrm{OB}}-\lambda \overrightarrow{\mathrm{OA}})$ for some $\lambda>0$. If $|\overrightarrow{\mathrm{OB}} \times \overrightarrow{\mathrm{OC}}|=\frac{9}{2}$, then which of the following statements is (are) TRUE?

(A) Projection of $\overrightarrow{\mathrm{OC}}$ on $\overrightarrow{\mathrm{OA}}$ is $-\frac{3}{2}$; (B) Area of the triangle OAB is $\frac{9}{2}$; (C) Area of the triangle ABC is $\frac{9}{2}$

(D) The acute angle between the diagonals of the parallelogram with adjacent sides $\overrightarrow{\mathrm{OA}}$ and $\overrightarrow{\mathrm{OC}}$ is $\frac{\pi}{3}$

JEE Advanced · 2021 · 03 Oct 2021 · Paper 2 · Vector Algebra

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Let O be the origin and

OA = 2 \hat{i} + 2 \hat{j} + \hat{k}, OB = \hat{i} - 2 \hat{j} + 2 \hat{k}

and

OC = \frac{1}{2} (OB - λ OA)

for some

λ > 0

. If

∣ OB \times OC ∣ = \frac{9}{2}

, then which of the following statements is (are) TRUE?

(A)Projection of $OC$ on $OA$ is $- \frac{3}{2}$ ✓ Correct
(B)Area of the triangle OAB is $\frac{9}{2}$ ✓ Correct
(C)Area of the triangle ABC is $\frac{9}{2}$ ✓ Correct
(D)The acute angle between the diagonals of the parallelogram with adjacent sides $OA$ and $OC$ is $\frac{π}{3}$

Answer: (A), (B), (C)

Solution

OB \times OC = \frac{1}{2} OB \times (OB - λ OA) = \frac{λ}{2} (OA \times OB)

.

Note that

OA \cdot OB = 2 (1) + 2 (- 2) + 1 (2) = 0

, so OA and OB are perpendicular. Thus

∣ OA \times OB ∣ = ∣ OA ∣∣ OB ∣ = 3 \times 3 = 9

.

From

∣ OB \times OC ∣ = \frac{9}{2}

\frac{λ}{2} \times 9 = \frac{9}{2}

, so

λ = 1

.

Thus

OC = \frac{1}{2} (OB - OA) = \frac{1}{2} AB

, meaning C is the midpoint of AB.

(A) Projection of

OC

OA

\frac{OC \cdot OA}{∣ OA ∣} = \frac{- 3/2 - 3/2 + 1/2}{3} = - \frac{3}{2}

. Correct.

(B) Area of triangle OAB =

\frac{1}{2} ∣ OA \times OB ∣ = \frac{9}{2}

. Correct.

(C) Since C is the midpoint of AB, triangle ABC has area

\frac{1}{2} \times

area of OAB

= \frac{9}{2}

. Correct.

(D) The acute angle between diagonals is

tan^{- 1} (2)

, not

\frac{π}{3}

. Incorrect.

Correct answers: (A), (B), (C).

JEE Advanced · 2021 · 03 Oct 2021 · Paper 2 · One or more correct options

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JEE Advanced · 2021 · 03 Oct 2021 · Paper 2 · Vector Algebra

Solution

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