Question:medium
For three non–coplanar vectors \(\vec a,\vec b,\vec c\), if
\[
(\vec b+\vec c)\cdot\big((\vec c+\vec a)\times(\vec a+\vec b)\big)
= \alpha \,[\vec a\,\vec b\,\vec c]
\]
and
\[
(\vec a+\vec b)\cdot\big((\vec b+\vec c)\times(\vec a+\vec b+\vec c)\big)
= \beta \,[\vec a\,\vec b\,\vec c],
\]
then \(\alpha+\beta\) is equal to:
For three non–coplanar vectors \(\vec a,\vec b,\vec c\), if
\[
(\vec b+\vec c)\cdot\big((\vec c+\vec a)\times(\vec a+\vec b)\big)
= \alpha \,[\vec a\,\vec b\,\vec c]
\]
and
\[
(\vec a+\vec b)\cdot\big((\vec b+\vec c)\times(\vec a+\vec b+\vec c)\big)
= \beta \,[\vec a\,\vec b\,\vec c],
\]
then \(\alpha+\beta\) is equal to:
Show Hint
Use the multilinearity and cyclic properties of the scalar triple product.
Most expanded terms vanish due to repetition of vectors.
Updated On: Mar 5, 2026
- \(-3\)
- \(-1\)
- \(1\)
- \(3\)
Show Solution
The Correct Option is D
Solution and Explanation
Was this answer helpful?
0
Top Questions on Vectors
- Vector of magnitude 3 making equal angles with $x$ and $y$ axes and perpendicular to $z$ axis is
- Let \(\alpha,\beta,\gamma\ (0<\alpha,\beta,\gamma<\tfrac{\pi}{2})\) be the angles between non–zero vectors \(\vec a\) and \(\vec b\), \(\vec b\) and \(\vec c\), \(\vec c\) and \(\vec a\) respectively. If \(\theta\) is the angle that the vector \(\vec a\) makes with the plane containing \(\vec b\) and \(\vec c\), then:
- JEE Main - 2026
- Mathematics
- Vectors
- If \( \theta \) is the angle between two vectors \( \vec{a} \) and \( \vec{b} \) such that \( |\vec{a}| = 7, |\vec{b}| = 1 \) and \[ |\vec{a} \times \vec{b}|^2 = k^2 - (\vec{a} - \vec{b})^2, \] then the values of \( k \) and \( \theta \) are:
- Let \( \vec{a}, \vec{b}, \vec{c} \) be unit vectors. Suppose \( \vec{a}\cdot\vec{b} = \vec{a}\cdot\vec{c} = 0 \) and the angle between \( \vec{b} \) and \( \vec{c} \) is \( \frac{\pi}{6} \). Then \( \vec{a} \) is:
- Want to practice more? Try solving extra ecology questions todayView All Questions
Questions Asked in JEE Main exam
In an equilateral prism the path of a ray is shown in the figure. Determine the refractive index of the prism.
- JEE Main - 2026
- Ray optics and optical instruments
