Let \(\vec{a}=9 \hat{i}-13 \hat{j}+25 \hat{k}, \vec{b}=3 \hat{i}+7 \hat{j}-13 \hat{k}\) and \(\overrightarrow{\mathrm{c}}=17 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+\hat{\mathrm{k}}\) be three given vectors. If \(\overrightarrow{\mathrm{r}}\) is a vector such that \(\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{a}}=(\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}) \times \overrightarrow{\mathrm{a}}\) and \(\overrightarrow{\mathrm{r}} \cdot(\overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{c}})=0\), then \(\frac{|593 \vec{r}+67 \vec{a}|^{2}}{(593)^{2}}\) is equal to ______ .

JEE Main · 2024 · 08 Apr 2024 · Shift 1 · Vector Algebra

Let

a = 9 \hat{i} - 13 \hat{j} + 25 \hat{k}, b = 3 \hat{i} + 7 \hat{j} - 13 \hat{k}

and

c = 17 \hat{i} - 2 \hat{j} + \hat{k}

be three given vectors. If

r

is a vector such that

r \times a = (b + c) \times a

and

r \cdot (b - c) = 0

, then

\frac{∣593 r + 67 a ∣ ^{2}}{( 593 ) ^{2}}

is equal to ______ .

Answer: 569

From

r \times a = (b + c) \times a

r = λ a + b + c

. Imposing

r \cdot (b - c) = 0

gives

λ = \frac{( c - b ) \cdot ( b + c )}{( b - c ) \cdot a} = \frac{- 67}{593}

. Then

593 r + 67 a = 593 (b + c)

, so

\frac{∣593 r + 67 a ∣ ^{2}}{59 3 ^{2}} = ∣ b + c ∣^{2} = ∣20 \hat{i} + 5 \hat{j} - 12 \hat{k} ∣^{2} = 400 + 25 + 144 = 569

JEE Main · 2024 · 08 Apr 2024 · Shift 1 · Integer answer