CBSE 2023 · Region 1 · Set 1 · Q23 · 2 marks
Position vectors of the points $\displaystyle \mathrm{A}, \mathrm{B}$ and C as shown in the figure below are $\displaystyle \overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}$ and $\displaystyle \overrightarrow{\mathrm{c}}$ respectively.
If $\displaystyle \overrightarrow{\mathrm{AC}}=\frac{5}{4} \overrightarrow{\mathrm{AB}}$, express $\displaystyle \overrightarrow{\mathrm{c}}$ in terms of $\displaystyle \overrightarrow{\mathrm{a}}$ and $\displaystyle \overrightarrow{\mathrm{b}}$.Check whether the lines given by equations $\displaystyle \mathrm{x}=2 \lambda+2, \mathrm{y}=7 \lambda+1$, $\displaystyle \mathrm{z}=-3 \lambda-3$ and $\displaystyle \mathrm{x}=-\mu-2, \mathrm{y}=2 \mu+8, \mathrm{z}=4 \mu+5$ are perpendicular to each other or not.
Position vectors of the points $\displaystyle \mathrm{A}, \mathrm{B}$ and C as shown in the figure below are $\displaystyle \overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}$ and $\displaystyle \overrightarrow{\mathrm{c}}$ respectively.
If $\displaystyle \overrightarrow{\mathrm{AC}}=\frac{5}{4} \overrightarrow{\mathrm{AB}}$, express $\displaystyle \overrightarrow{\mathrm{c}}$ in terms of $\displaystyle \overrightarrow{\mathrm{a}}$ and $\displaystyle \overrightarrow{\mathrm{b}}$.
Check whether the lines given by equations $\displaystyle \mathrm{x}=2 \lambda+2, \mathrm{y}=7 \lambda+1$, $\displaystyle \mathrm{z}=-3 \lambda-3$ and $\displaystyle \mathrm{x}=-\mu-2, \mathrm{y}=2 \mu+8, \mathrm{z}=4 \mu+5$ are perpendicular to each other or not.
Marking-scheme solution
(a)
According to question, $\displaystyle \vec{\mathrm{c}}-\vec{\mathrm{a}}=\frac{5}{4}(\vec{\mathrm{b}}-\vec{\mathrm{a}})$
$$\therefore \vec{\mathrm{c}}=\frac{$\displaystyle 5$ \vec{\mathrm{b}}}{$\displaystyle 4$}-\frac{\vec{\mathrm{a}}}{$\displaystyle 4$}
$$OR
(b) D.r.s. of lines are < $\displaystyle 2$, $\displaystyle 7$, -$\displaystyle 3$ > and <-$\displaystyle 1$, $\displaystyle 2$, $\displaystyle 4>$
Now 2. $\displaystyle -1+7 \cdot 2+-3 \cdot 4=0$
∴ given lines are perpendicular
Vector AlgebraProduct of Two VectorsApplyvery_short_answermedium
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CBSE Class 12 Mathematics past-paper question from the 2023board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.