Find the shortest distance between the lines r -2 -k-k- and r -2 -k-2 -2 k-

The given equations are r= î+2 ĵ+k̂+λ (î ĵ+k̂) and r = 2 î ĵ k̂+μ (2 î+ĵ+2k̂) which is of the form r = a_1 + λ b_1 and r = a_2 + μ b_2 a_1 =î + 2 ĵ + k̂, ...

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

Byju's Answer

Standard XII

Mathematics

Shortest Distance between Two Skew Lines

Find the shor...

Question

Find the shortest distance between the lines
$r = (^i + 2^j +^k) + λ (^i -^j +^k) a n d r = (2^i -^j -^k) + μ (2^i +^j + 2^k)$

Open in App

Solution

The given equations are $r =^i + 2^j +^k + λ (^i -^j +^k)$
and $r = 2^i -^j -^k + μ (2^i +^j + 2^k)$
which is of the form $r = a_{1} + λ b_{1} a n d r = a_{2} + μ b_{2}$
$a_{1} =^i + 2^j +^k, b_{1} =^i -^j +^k$
and $a_{2} = 2^i -^j -^k, b_{2} = 2^i +^j + 2^k$
Now, $a_{2} - a_{1} = (2^i -^j -^k) - (^i + 2^j +^k) =^i - 3^j - 2^k$
and $b_{1} \times b_{2} = ∣ ∣ ∣ ∣ ∣ \begin{matrix} ^i & ^j & ^k 1 & - 1 & 1 2 & 1 & 2 \end{matrix} ∣ ∣ ∣ ∣ ∣ =^i (- 2 - 1) -^j (2 - 2) +^k (1 + 2) = - 3^i + 3^k$
$\Rightarrow | b_{1} \times b_{2} | = \sqrt{(- 3)^{2} + (3)^{2}} = \sqrt{9 \times 9} = \sqrt{18} = 3 \sqrt{2}$
Substituting all the values in equation, wa obtain
Shortest distance $= ∣ ∣ \frac{(b_{1} \times b_{2}) . (a_{2} - a_{1})}{| b_{1} \times b_{2} |} ∣ ∣$
$= \frac{| (- 3^i + 3^k) . (^i - 3^j - 2^k) |}{3 \sqrt{2}} = \frac{| (- 3) \times 1 + 0 \times (- 3) + 3 \times (- 2) |}{3 \sqrt{2}} = \frac{9}{3 \sqrt{2}} = \frac{3 \times \sqrt{2}}{\sqrt{2} \times \sqrt{2}} = \frac{3 \sqrt{2}}{2} u n i t s$
Therefore, the shortest distance between the two lines is $\frac{3 \sqrt{2}}{2}$ units.
Note: The two lines should be parallel and non - interceping, then we can only fetermined the shortest distance.

Suggest Corrections

Similar questions

Q. Find the shortest distance between the lines.

\to r =^i + 2^j +^k + λ (^i -^j +^k)

\to r = 2^i -^j -^k + μ (2^i +^j + 2^k)

Q. The distance (in units) between the parallel lines

\to r =^i + 2^j + 3^k + λ (^i -^j +^k)

and

\to r = 2^i -^j -^k + μ (^i -^j +^k)

Q. Find the shortest distance between the following pairs of parallel lines.

\to r = (^i + 2^j + 3^k) + λ (^i -^j +^k)

and

\to r = (2^i -^j -^k) + μ (-^i +^j -^k)

Q. Find the shortest distance between the lines

\to r =^i +^j + λ (2^i -^j +^k)

\to r = 2^i +^j -^k + μ (3^i - 5^j + 2^k)

Q. Find the shortest distance between the lines

l_{1}

and

l_{2}

whose vector equations are

\to r = 2^i +^j +^k + λ (^i - 2^j +^k)

and

\to r =^i - 3^j -^k + λ (5^i + 2^j -^k)

Join BYJU'S Learning Program

Find the shortest distance between the lines r=(^i+2^j+^k)+λ(^i−^j+^k) and r=(2^i−^j−^k)+μ(2^i+^j+2^k)

Find the shortest distance between the lines
$r = (^i + 2^j +^k) + λ (^i -^j +^k) a n d r = (2^i -^j -^k) + μ (2^i +^j + 2^k)$