Let a -2 -3 -6 k- b -2 -3 -6 k- and c -2 -3 -6 k- Let a 1 be the projection of a on b and a 2 be the projection of a 1on c- Then- which of the following is true-A- a- and a-2 are collinearB- a 1 and c are collinearC- a - a 1 and c are coplanarD- a - a 1 and b are coplanar

The correct option is D a, nbsp;a_1 and b are coplanara_1=[(2î+3ĵ 6k̂) ·(2î 3ĵ+6k̂)7] (2î 3ĵ+6k̂)7 ⇒ a_1= 4149(2î 3ĵ+6k̂) a, a_1, and b are coplanar because ...

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Byju's Answer

Standard XII

Mathematics

Test for Coplanarity

Let a =2 î+3 ...

Question

Let $\to a = 2^i + 3^j - 6^k, \to b = 2^i - 3^j + 6^k$ and $\to c = - 2^i + 3^j + 6^k$ . Let ${\to a}_{1}$ be the projection of $\to a$ on $\to b$ and ${\to a}_{2}$ be the projection of ${\to a}_{1}$ on $\to c$ . Then, which of the following is true?

\to a

and

{\to a}_{2}

are collinear

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{\to a}_{1}

and

\to c

are collinear

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\to a, {\to a}_{1}

and

\to b

are coplanar

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\to a, {\to a}_{1}

and

\to c

are coplanar

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Solution

The correct option is C $\to a, {\to a}_{1}$ and $\to b$ are coplanar
$a_{1} = [(2^i + 3^j - 6^k) \cdot \frac{(2^i - 3^j + 6^k)}{7}] \frac{(2^i - 3^j + 6^k)}{7}$
$\Rightarrow a_{1} = - \frac{41}{49} (2^i - 3^j + 6^k)$
$\to a, {\to a}_{1},$ and $\to b$ are coplanar because ${\to a}_{1}$ and $\to b$ are collinear.

Suggest Corrections

Similar questions

Q. Let

\to a = 2^i + 3^j - 6^k, \to b = 2^i - 3^j + 6^k

and

\to c = - 2^i + 3^j + 6^k

. Let

{\to a}_{1}

be the projection of

\to a

\to b

and

{\to a}_{2}

be the projection of

{\to a}_{1}

\to c

. Then,

{\to a}_{1} . \to b

is equal to

Q. Let

\to a = 2^i + 3^j - 6^k, \to b = 2^i - 3^j + 6^k

and

\to c = - 2^i + 3^j + 6^k

. Let

{\to a}_{1}

be the projection of

\to a

\to b

and

{\to a}_{2}

be the projection of

{\to a}_{1}

\to c

. Then,

{\to a}_{1} . \to b

is equal to

Q. Let

\to a = 2^i + 3^j - 6^k, \to b = 2^i - 3^j + 6^k

and

\to c = - 2^i + 3^j + 6^k

. Let

{\to a}_{1}

be the projection of

\to a

\to b

and

{\to a}_{2}

be the projection of

{\to a}_{1}

\to c

. Then,

{\to a}_{2}

is equal to

Q. Given two vectors

a = 2^i - 3^j + 6^k

b = - 2^i + 2^j - 1^k

and

λ

= ratio of the projection of

a

b

and the projection of

b

a,

then the value of

λ

Q. Position vectors of A, B, C are given by

\to a, \to b, \to c

where

\to a \times \to b + \to b \times \to c + \to c \times \to a = 0

. If

\to A C = 2^i - 3^j + 6^k

find

\to B C

if BC

= 14

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Let →a=2^i+3^j−6^k, →b=2^i−3^j+6^k and →c=−2^i+3^j+6^k. Let →a1 be the projection of →a on →b and →a2 be the projection of →a1 on →c. Then, which of the following is true?

The correct option is C →a, →a1 and →b are coplanara1=[(2^i+3^j−6^k)⋅(2^i−3^j+6^k)7](2^i−3^j+6^k)7 ⇒a1=−4149(2^i−3^j+6^k) →a, →a1, and →b are coplanar because →a1 and →b are collinear.

Let $\to a = 2^i + 3^j - 6^k, \to b = 2^i - 3^j + 6^k$ and $\to c = - 2^i + 3^j + 6^k$ . Let ${\to a}_{1}$ be the projection of $\to a$ on $\to b$ and ${\to a}_{2}$ be the projection of ${\to a}_{1}$ on $\to c$ . Then, which of the following is true?