SATHEE: Vectors 5 Question 15

Vectors 5 Question 15

15. If vectors $\vec{a}, \vec{b}, \vec{c}$ are coplanar, then show that

$| \begin{array}{ccc} \vec{a} & \vec{b} & \vec{c} \\ \vec{a} \cdot \vec{a} & \vec{a} \cdot \vec{b} & \vec{a} \cdot \vec{c} \\ \vec{b} \cdot \vec{a} & \vec{b} \cdot \vec{b} & \vec{b} \cdot \vec{c} \end{array} | = \vec{0}$

(1989, 2M)

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Solution:

Given that, $\vec{a} \vec{b} \vec{c}$ are coplanar vectors.

$∴$ There exists scalars $x, y, z$ not all zero, such that

$x \vec{a} + y \vec{b} + z \vec{c} = 0$

Taking dot with $\vec{a}$ and $\vec{b}$ respectively, we get

$\begin{aligned} x (\vec{a} \cdot \vec{a}) + y (\vec{a} \cdot \vec{b}) + z (\vec{a} \cdot \vec{c}) = 0 \\ and x (\vec{a} \cdot \vec{b}) + y (\vec{b} \cdot \vec{b}) + z (\vec{c} \cdot \vec{b}) = 0 \end{aligned}$

Since, Eqs. (i), (ii) and (iii) represent homogeneous equations with $(x, y, z) \neq (0, 0, 0)$ .

$\Rightarrow$ Non-trivial solutions

$∴ Δ = 0 \Rightarrow | \begin{array}{ccc} \vec{a} & \vec{b} & \vec{c} \vec{a} \cdot \vec{a} & \vec{a} \cdot \vec{b} & \vec{a} \cdot \vec{c} \vec{b} \cdot \vec{b} & \vec{b} \cdot \vec{b} & \vec{b} \cdot \vec{c} \end{array} | = \vec{0}$

Vectors 5 Question 15

15. If vectors $\vec{a}, \vec{b}, \vec{c}$ are coplanar, then show that

Solution:

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Vectors 5 Question 15

15. If vectors a→,b→,c→ are coplanar, then show that

Solution:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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15. If vectors $\vec{a}, \vec{b}, \vec{c}$ are coplanar, then show that