Consider three vectors A - - - - - 2k- B-k- and C-2-3j-4k- A vector X of the form -A - -B - and - are numbers in perpendicular to C- The ratio of - and - is

The correct option is A 1:1 αA⃗=αî+αĵ 2αk̂ βB⃗=βî βĵ+βk̂ Hence (αA⃗+βB⃗).C⃗=[(α+β)î+(α 3β)ĵ+( 2α+4β)k̂].(2î 3ĵ+4k̂)=0 (Since they are perpe ...

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

Byju's Answer

Standard XII

Physics

Work Done as Dot Product

Consider thre...

Question

Consider three vectors $\to A =^i +^j - 2^k, \to B =^i -^j +^k$ and $\to C = 2^i - 3 j + 4^k . A$ vector $\to X$ of the form $α \to A + β \to B$ ( $α$ and $β$ are numbers) in perpendicular to $\to C$ . The ratio of $α$ and $β$ is

1 : 1

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

2 : 1

No worries! We‘ve got your back. Try BYJU‘S free classes today!

- 1 : 1

No worries! We‘ve got your back. Try BYJU‘S free classes today!

3 : 1

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is A $1 : 1$
$α \to A = α^i + α^j - 2 α^k$
$β \to B = β^i - β^j + β^k$
Hence $(α \to A + β \to B) . \to C = [(α + β)^i + (α - 3 β)^j + (- 2 α + 4 β)^k] . (2^i - 3^j + 4^k) = 0$ (Since they are perpendicular)
$⟹ 2 (α + β) - 3 (α - 3 β) + 4 (- 2 α + 4 β) = 0$
$⟹ α = β$
$⟹ \frac{α}{β} = 1$

Suggest Corrections

Similar questions

Q. Consider three vectors

\to A =^i +^j - 2^k, \to B =^i -^j +^k

and

\to C = 2^i - 3^j + 4^k

. A vector

\to X

of the form

α \to A + β \to B

(

α

and

β

are numbers) is perpendicular to

\to C

. The ratio of

α

and

β

is:

Q. If

\to a =^i +^j +^k,^b = 4^i + 3^j + 4^k

and

\to c =^i + α^j + β^k

are linearly dependent vectors and

| \to c | = \sqrt{3},

then

Q. Let

\to a = 2^i +^j -^k

and

\to b =^i + 2^j +^k

be two vectors. Consider a vector

\to c = α \to a + β \to b, α, β \in R .

If the projection of

\to c

on the vector

(\to a + \to b)

3 \sqrt{2},

then the minimum value of

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

equals

Q. Let

\to a = 2^i +^j -^k

and

\to b =^i + 2^j +^k

be two vectors. Consider a vector

\to c = α \to a + β \to b, α, β \in R .

If the projection of

\to c

on the vector

(\to a + \to b)

3 \sqrt{2},

then the minimum value of

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

equals

Q. If the vectors

α^i +^j +^k,^i + β^j +^k,^i +^j + λ^k (α, β, γ \neq 1)

are coplanar, then the value of

\frac{1}{1 - α} + \frac{1}{1 + β} + \frac{1}{1 - γ}

is.

Join BYJU'S Learning Program

Consider three vectors →A=^i+^j−2^k,→B=^i−^j+^k and →C=2^i−3j+4^k.A vector →X of the form α→A+β→B (α and β are numbers) in perpendicular to →C. The ratio of α and β is

The correct option is A 1:1α→A=α^i+α^j−2α^kβ→B=β^i−β^j+β^kHence (α→A+β→B).→C=[(α+β)^i+(α−3β)^j+(−2α+4β)^k].(2^i−3^j+4^k)=0(Since they are perpendicular)⟹2(α+β)−3(α−3β)+4(−2α+4β)=0⟹α=β⟹αβ=1

Consider three vectors $\to A =^i +^j - 2^k, \to B =^i -^j +^k$ and $\to C = 2^i - 3 j + 4^k . A$ vector $\to X$ of the form $α \to A + β \to B$ ( $α$ and $β$ are numbers) in perpendicular to $\to C$ . The ratio of $α$ and $β$ is