CameraIcon

SearchIcon

MyQuestionIcon

1

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Selecting Consecutive Terms in A.P

If a , → b , ...

Question

If $\vec{a,} \vec{b,} \vec{c}$ are three non-coplanar vectors, then $(\vec{a} + \vec{b} + \vec{c}) . [(\vec{a} + \vec{b}) \times (\vec{a} + \vec{c})]$ equals

(a) 0

(b) $[\vec{a} \vec{b} \vec{c}]$

(c) $2 [\vec{a} \vec{b} \vec{c}]$

(d) $- [\vec{a} \vec{b} \vec{c}]$

Open in App

Solution

$(d) - [\vec{a} \vec{b} \vec{c}] We have (\vec{a} + \vec{b} + \vec{c}) . [(\vec{a} + \vec{b}) \times (\vec{a} + \vec{c})] = (\vec{a} + \vec{b} + \vec{c}) . [(\vec{a} + \vec{b}) \times \vec{a} + (\vec{a} + \vec{b}) \times \vec{c}] (By definition of cross poduct) = (\vec{a} + \vec{b} + \vec{c}) . [\vec{a} \times \vec{a} + \vec{b} \times \vec{a} + \vec{a} \times \vec{c} + \vec{b} \times \vec{c}] = (\vec{a} + \vec{b} + \vec{c}) . [0 + \vec{b} \times \vec{a} + \vec{a} \times \vec{c} + \vec{b} \times \vec{c}] = \vec{a} . (\vec{b} \times \vec{a}) + \vec{a} . (\vec{a} \times \vec{c}) + \vec{a} . (\vec{b} \times \vec{c}) + \vec{b} . (\vec{b} \times \vec{a}) + \vec{b} . (\vec{a} \times \vec{c}) + \vec{b} . (\vec{b} \times \vec{c}) + \vec{c} . (\vec{b} \times \vec{a}) + \vec{c} . (\vec{a} \times \vec{c}) + \vec{c} . (\vec{b} \times \vec{c}) = [\vec{a} \vec{b} \vec{a}] + [\vec{a} \vec{a} \vec{c}] + [\vec{a} \vec{b} \vec{c}] + [\vec{b} \vec{b} \vec{a}] + [\vec{b} \vec{a} \vec{c}] + [\vec{b} \vec{b} \vec{c}] + [\vec{c} \vec{b} \vec{a}] + [\vec{c} \vec{a} \vec{c}] + [\vec{c} \vec{b} \vec{c}] = 0 + 0 + [\vec{a} \vec{b} \vec{c}] + 0 + [\vec{b} \vec{a} \vec{c}] + 0 + [\vec{c} \vec{b} \vec{a}] + 0 + 0 = [\vec{a} \vec{b} \vec{c}] - [\vec{a} \vec{b} \vec{c}] - [\vec{a} \vec{b} \vec{c}] (∵ [\vec{b} \vec{a} \vec{c}] = - [\vec{c} \vec{a} \vec{b}], [\vec{b} \vec{a} \vec{c}] = - [\vec{a} \vec{b} \vec{c}]) = - [\vec{a} \vec{b} \vec{c}]$

Suggest Corrections

thumbs-up

0

Similar questions

\to a, \to b

\to c

are three non – coplanar vectors, then

(\to a + \to b + \to c) . ((\to a + \to b) \times (\to a + \to c))

\vec{a}, \vec{b}, \vec{c}

are non-coplanar vectors, then

\frac{(\vec{b} \times \vec{c}), (\vec{a} + \vec{b} + \vec{c})}{[\vec{a} \vec{b} \vec{c}]}

= _______________.

\vec{a,} \vec{b,} \vec{c}

are non-coplanar vectors, then

\frac{\vec{a} \cdot (\vec{b} \times \vec{c})}{(\vec{c} \times \vec{a}) \cdot \vec{b}} + \frac{\vec{b} \cdot (\vec{a} \times \vec{c})}{\vec{c} \cdot (\vec{a} \times \vec{b})}

is equal to
(a) 0
(b) 2
(c) 1
(d) none of these

If $\to a, \to b$ and $\to c$ are three non-coplanar vectors, then $(\to a + \to b + \to c) \cdot [(\to a + \to b) \times (\to a + \to c)]$ equals

If $\to a, \to b$ and $\to c$ are three non-coplanar vectors, then $(\to a + \to b + \to c) \cdot [(\to a + \to b) \times (\to a + \to c)]$ equals

View More

Join BYJU'S Learning Program

similar_icon

Related Videos

lock

Arithmetic Progression

MATHEMATICS

Watch in App

explore_more_icon

Explore more

Selecting Consecutive Terms in A.P

Standard XII Mathematics

Join BYJU'S Learning Program

CrossIcon