# For Non-zero Vectors → a , → B and → C the Relation - Mathematics

MCQ
Sum

For non-zero vectors $\vec{a,} \vec{b} \text { and }\vec{c}$ the relation $\left| \left( \vec{a} \times \vec{b} \right) \cdot \vec{c} \right| = \left| \vec{a} \right| \left| \vec{b} \right| \left| \vec{c} \right|$ holds good, if

#### Options

• $\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{c} = 0$

• $\vec{a} \cdot \vec{b} = 0 = \vec{c} \cdot \vec{a}$

• $\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{c} = \vec{c} \cdot \vec{a} = 0$

• $\vec{b} \cdot \vec{c} = \vec{c} \cdot \vec{a} = 0$

#### Solution

$\vec{a} . \vec{b} = \vec{b} . \vec{c} = \vec{c} . \vec{a} = 0$

We have

$\left| \left( \vec{a} \times \vec{b} \right) . \vec{c} \right|$

$= \left| \left( \vec{a} \times \vec{b} \right) \right| \left| \vec{c} \right|\left| cos\theta \right|$

$= \left| \left( \vec{a} \times \vec{b} \right) \right| \left| \vec{c} \right| \left( \text { If } \theta = 0^\circ \text { or } 180^\circ , \text { i . e . vectors } \vec{a} \times \vec{b} \text { and }\vec {c}\text { are parallel } \right)$

$= \left| \left( \left| \vec{a} \right|\left| \vec{b} \right| \sin \alpha \right) \right|\left| \vec{c} \right|$

$= \left| \vec{a} \right|\left| \vec{b} \right|\left| \vec{c} \right| \left( \text { If } \alpha = 90^\circ,\text { i . e . vectors }\vec{a}\text { and } \vec{b} \text { are perpendicular } \right)$

$\therefore \left| \left( \vec{a} \times \vec{b} \right) . \vec{c} \right| = \left| \vec{a} \right|\left| \vec{b} \right|\left| \vec{c} \right| \left(\text { If vectors } \vec{a} , \vec{b} , \vec{c} \text { are perpendicular to each other } \right)$

$\text { Thus, the relation } \left| \left( \vec{a} \times \vec{b} \right) . \vec{c} \right| = \left| \vec{a} \right|\left| \vec{b} \right|\left| \vec{c} \right|\text { holds good if } \vec{a} . \vec{b} = 0 , \vec{b} . \vec{c} = 0 \text { and } \vec{c} . \vec{a} = 0 .$

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#### APPEARS IN

RD Sharma Class 12 Maths
Chapter 26 Scalar Triple Product
MCQ | Q 12 | Page 19