CBSE 2026 · Region 5 · Set 1 · Q30 · 3 marks
Let three toys A, B and C be placed in the same straight line. If the position vectors of A, B and C are $\displaystyle 55 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}, 5 \hat{\mathrm{i}}+8 \hat{\mathrm{j}}$ and $\displaystyle \mathrm{a} \hat{\mathrm{i}}-52 \hat{\mathrm{j}}$ respectively, find the value of 'a'.If $\displaystyle \overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}$ and $\displaystyle \overrightarrow{\mathrm{c}}$ are unit vectors, then prove that \[|\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}|^{2}+|\overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{c}}|^{2}+|\overrightarrow{\mathrm{c}}-\overrightarrow{\mathrm{a}}|^{2} \leq 9 . \]
Let three toys A, B and C be placed in the same straight line. If the position vectors of A, B and C are $\displaystyle 55 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}, 5 \hat{\mathrm{i}}+8 \hat{\mathrm{j}}$ and $\displaystyle \mathrm{a} \hat{\mathrm{i}}-52 \hat{\mathrm{j}}$ respectively, find the value of 'a'.
If $\displaystyle \overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}$ and $\displaystyle \overrightarrow{\mathrm{c}}$ are unit vectors, then prove that \[|\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}|^{2}+|\overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{c}}|^{2}+|\overrightarrow{\mathrm{c}}-\overrightarrow{\mathrm{a}}|^{2} \leq 9 . \]
Marking-scheme solution
$\displaystyle \overrightarrow{AB}=-50 \hat{i}+10 \hat{j}$ and $\displaystyle \overrightarrow{BC}=(a-5) \hat{i}-60 \hat{j}$
As $\displaystyle \overrightarrow{AB}$ and $\displaystyle \overrightarrow{BC}$ are collinear vectors
$\displaystyle \therefore \dfrac{-50}{a-5}=\dfrac{10}{-60}$
$\displaystyle a=305$
$\displaystyle |\vec{a}|=|\vec{b}|=|\vec{c}|=1$.
Now, consider
$\displaystyle |\vec{a}-\vec{b}|^{2}+|\vec{b}-\vec{c}|^{2}+|\vec{c}-\vec{a}|^{2}$
$\displaystyle =2|\vec{a}|^{2}+2|\vec{b}|^{2}+2|\vec{c}|^{2}-2 \vec{a} \cdot \vec{b}-2 \vec{b} \cdot \vec{c}-2 \vec{c} \cdot \vec{a}$
$\displaystyle =3|\vec{a}|^{2}+3|\vec{b}|^{2}+3|\vec{c}|^{2}-\left(|\vec{a}|^{2}+|\vec{b}|^{2}+|\vec{c}|^{2}+2 \vec{a} \cdot \vec{b}+2 \vec{b} \cdot \vec{c}+2 \vec{c} \cdot \vec{a}\right)$
$\displaystyle =9-|\vec{a}+\vec{b}+\vec{c}|^{2} \leq 9$
$\displaystyle \therefore|\vec{a}-\vec{b}|^{2}+|\vec{b}-\vec{c}|^{2}+|\vec{c}-\vec{a}|^{2} \leq 9$
Vector AlgebraAddition of VectorsApplyshort_answermedium
More from Vector Algebra
- 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…2025 · asked 3×
- The position vectors of points P and Q are p and q respectively. The point R divides line segment PQ in the…2024 · asked 3×
- If a+ b= i and a=2 i-2 j+2 k, then b equals:2023 · asked 3×
- A vector a makes equal angles with all the three axes. If the magnitude of the vector is 5 √3 units, then…2025 · asked 3×
- The vector with terminal point A(2,-3,5) and initial point B(3,-4,7) is:2024 · asked 3×
- Find the position vector of point C which divides the line segment joining points A and B having position…2024 · asked 3×
- If a =2 and -3 ≤ k ≤ 2, then k a ∈:2024 · asked 3×
- Unit vector along PQ, where coordinates of P and Q respectively are (2,1,-1) and (4,4,-7), is2023 · asked 3×
CBSE Class 12 Mathematics past-paper question from the 2026board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.