✓ Board-verified
Mathematics · 2022 · 3 marks
CBSE 2022 · Region 3 · Set 2 · Q7
If a, b, c are three vectors such that a · b = a · c and a × b = a × c, a ≠ 0, then show that b = c. OR If | a |=3,| b |=5,| c |=4 and a + b + c = 0,…
If $\displaystyle \overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$ are three vectors such that $\displaystyle \overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}=\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{c}}$ and $\displaystyle \overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}=$ $\displaystyle \overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{c}}, \overrightarrow{\mathrm{a}} \neq 0$, then show that $\displaystyle \overrightarrow{\mathrm{b}}=\overrightarrow{\mathrm{c}}$.
OR If $\displaystyle |\overrightarrow{\mathrm{a}}|=3,|\overrightarrow{\mathrm{~b}}|=5,|\overrightarrow{\mathrm{c}}|=4$ and $\displaystyle \overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}=\overrightarrow{0}$, then find the value of $\displaystyle (\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{c}} \cdot \overrightarrow{\mathrm{a}})$.
Marking-scheme solution
$\displaystyle \vec{a} \cdot \vec{b} = \vec{a} \cdot \vec{c}$
$\displaystyle \Rightarrow \vec{a} \cdot (\vec{b} - \vec{c}) = 0$
$\displaystyle \Rightarrow \vec{a} = 0,\ \vec{b} = \vec{c}$ or $\displaystyle \vec{a} \perp (\vec{b} - \vec{c})$
but $\displaystyle a \neq 0 \Rightarrow \vec{b} = \vec{c}$ or $\displaystyle \vec{a} \perp (\vec{b} - \vec{c})$
$\displaystyle \vec{a} \times \vec{b} = \vec{a} \times \vec{c}$
$\displaystyle \vec{a} \times (\vec{b} - \vec{c}) = 0$
$\displaystyle \vec{a} = 0,\ \vec{b} = \vec{c}$ or $\displaystyle \vec{a} \,||\, (\vec{b} - \vec{c})$
From ($\displaystyle 1$) and
$\displaystyle \vec{a} \neq 0$ given and $\displaystyle \vec{a}$ cannot be both $\displaystyle \perp$ and $\displaystyle \|$ to $\displaystyle (\vec{b} - \vec{c})$
$\displaystyle \therefore\ \vec{b} = \vec{c}$
$\displaystyle \vec{a} + \vec{b} + \vec{c} = 0$ gives $\displaystyle |\vec{a} + \vec{b} + \vec{c}|^2 = 0$
$\displaystyle \Rightarrow |\vec{a}|^2 + |\vec{b}|^2 + |\vec{c}| + 2\vec{a} \cdot \vec{b} + 2\vec{b} \cdot \vec{c} + 2\vec{c} \cdot \vec{a} = 0$
$\displaystyle \Rightarrow 9 + 25 + 16 + 2(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}) = 0$
$\displaystyle \Rightarrow \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} = -25$
All Vector Algebra questions2022 question paperNCERT solutions: Vector AlgebraPractice Vector Algebra →
More from Vector Algebra
- If a and b are two non-collinear vectors, then find x, such that α=(x-2) a+ b and β=(3+2 x) a-2 b are collinear.2025
- Three friends A, B and C move out from the same location O at the same time in three different directions to reach their destinations. They move out…2025
- If a+ b+ c= 0 such that | a|=3,| b|=5,| c|=7, then find the angle between a and b. OR If a and b are unit vectors inclined with each other at an…2025
- Find a vector of magnitude 4 units perpendicular to each of the vectors 2 i- j+ k and i+ j- k and hence verify your answer.2024
- The position vectors of points P and Q are p and q respectively. The point R divides line segment PQ in the ratio 3: 1 and S is the mid-point of line…2024
- If | a × b|=√3 and a · b=-3, then angle between a and b is2023
- Let p and q be two unit vectors and α be the angle between them. Then ( p+ q) will be a unit vector for what value of α?2025
- Using vectors, find the value of 'b' if the points A(-1, -1, 2), B(2, b, 5) and C(3, 11, 6) are collinear. Also, determine the ratio in which the…2022
CBSE Class 12 Mathematics past-paper question from the 2022 board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.