Advertisements
Advertisements
प्रश्न
Find λ for which the points A (3, 2, 1), B (4, λ, 5), C (4, 2, −2) and D (6, 5, −1) are coplanar.
Advertisements
उत्तर
\[\text { The points A, B, C and D will be coplanar iff any one of the following traces of vectors are coplanar: } \]
\[ \overrightarrow{AB} , \overrightarrow{AC} , \overrightarrow{AD} ; \overrightarrow{AB} , \overrightarrow{BC} , \overrightarrow{CD} ; \overrightarrow{BC} , \overrightarrow{BA} , \overrightarrow{BD} , \text { etc } . \]
\[\text {It is given that}\overrightarrow{AB } , \overrightarrow{ AC} , \overrightarrow{AD}\text{ are coplanar } . \]
\[\text { Thus, their scaler triple product } \left[ \overrightarrow{AB} \overrightarrow{AC} \overrightarrow{AD} \right] \text { is equal to zero } . \]
Now,
\[\text { Direction ratios of the } \overrightarrow{PQ} = \left(\text{ Direction ratios of vector Q } \right) - \left( \text {Direction ratios of the vector P} \right)\]
\[\text { Direction ratios of vector }| \overrightarrow{AB} = \left( 4 - 3 \right), \left( \lambda - 2 \right), \left( 5 - 1 \right), \text { i . e } . 1, \lambda - 2, 4\]
\[\text { Direction ratios of vector } \overrightarrow{AC} = \left( 4 - 3 \right), \left( 2 - 2 \right), \left( - 2 - 1 \right), \text { i . e } . 1, 0, - 3\]
\[\text { Direction ratios of vector } \overrightarrow{AD} = \left( 6 - 3 \right), \left( 5 - 2 \right), \left( - 1 - 1 \right), \text{ i . e }. 3, 3, - 2\]
\[ \therefore \left[ \overrightarrow{AB} \overrightarrow{AC} \overrightarrow{AD} \right] = \begin{vmatrix}1 & \lambda - 2 & 4 \\ 1 & 0 & - 3 \\ 3 & 3 & - 2\end{vmatrix} = 1\left[ 0 - \left( - 9 \right) \right] - \left( \lambda - 2 \right)\left[ - 2 - \left( - 9 \right) \right] + 4\left( 3 - 0 \right) = 0 \]
\[ \Rightarrow 7\lambda = 35\]
\[ \Rightarrow \lambda = 5\]
APPEARS IN
संबंधित प्रश्न
If A, B, C, D are (1, 1, 1), (2, 1, 3), (3, 2, 2), (3, 3, 4) respectively, then find the volume of parallelopiped with AB, AC and AD as the concurrent edges.
If `bar c = 3bara- 2bar b ` Prove that `[bar a bar b barc]=0`
Find the volume of a tetrahedron whose vertices are A(−1, 2, 3), B(3, −2, 1), C(2, 1, 3) and D(−1, −2, 4).
Let `veca = hati + hatj + hatk = hati` and `vecc = c_1veci + c_2hatj + c_3hatk` then
1) Let `c_1 = 1` and `c_2 = 2`, find `c_3` which makes `veca, vecb "and" vecc`coplanar
2) if `c_2 = -1` and `c_3 = 1`, show that no value of `c_1`can make `veca, vecb and vecc` coplanar
Give a condition that three vectors \[\vec{a}\], \[\vec{b}\] and \[\vec{c}\] form the three sides of a triangle. What are the other possibilities?
Evaluate the following:
\[\left[\hat{i}\hat{j}\hat{k} \right] + \left[ \hat{j}\hat{k}\hat {i} \right] + \left[ \hat{k}\hat{i} \hat{j} \right]\]
Evaluate the following:
\[\left[ 2 \hat{i}\hat{ j}\ \hat{k}\right] + \left[\hat{i}\hat{ k}\hat {j} \right] + \left[\hat{ k}\hat{ j} 2\hat{ i} \right]\]
Find \[\left[ \vec{a} \vec{b} \vec{c} \right]\] , when \[\vec{a} =\hat{ i} - 2 \hat{j} + 3 \hat{k} , \vec{b} = 2 \hat{i} + \hat{j} - \hat{k}\text{ and } \vec{c} = \hat{j} + \hat{k}\]
Find the volume of the parallelopiped whose coterminous edges are represented by the vector:
\[\vec{a} = \hat{i} + \hat{j} + \hat{k} , \vec{b} =\hat{ i} - \hat{j} + \hat{k} , \vec{c} = \hat{i} + 2 \hat{j} - \hat{k}\]
Find the value of λ for which the four points with position vectors
\[-\hat { j} - \hat {k} , 4 \hat {i} + 5 \hat {j} + \lambda \hat {k} , 3 \hat {i} + 9 \hat {j} + 4 \hat {k} \text { and } - 4 \hat {i} + 4 \hat {j} + 4 \hat{k}\]
Write the value of \[\left[ \hat {i} - \hat {j} \hat {j} - \hat {k} \hat {k} - \hat {i} \right] .\]
If the vectors (sec2 A) \[\hat {i} + \hat {j} + \hat {k} , \hat {i} + \left( \sec^2 B \right) \hat {j} + \hat {k} , \hat {i} + \hat {j} + \left( \sec^2 C \right) \hat {k}\] are coplanar, then find the value of cosec2 A + cosec2 B + cosec2 C.
If \[\left[ 3 \vec{a} + 7 \vec{b} \vec{c} \vec{d} \right] = \lambda\left[ \vec{a} \vec{c} \vec{d} \right] + \mu\left[ \vec{b} \vec{c} \vec{d} \right],\] then find the value of λ + μ.
If \[\vec{a}\] lies in the plane of vectors \[\vec{b} \text { and } \vec{c}\], then which of the following is correct?
If \[\left[ 2 \vec{a} + 4 \vec{b} \vec{c} \vec{d} \right] = \lambda\left[ \vec{a} \vec{c} \vec{d} \right] + \mu\left[ \vec{b} \vec{c} \vec{d} \right],\] then λ + μ =
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
\[\left( \vec{a} + \vec{b} \right) \cdot \left( \vec{b} + \vec{c} \right) \times \left( \vec{a} + \vec{b} + \vec{c} \right) =\]
Show that the vectors `hat (i) - 2 hat(j) + 3 hat (k), - 2 hat(i) + 3 hat(j) - 4 hat(k) " and " hat(i) - 3 hat(j) + 5 hat(k) ` are coplanar.
Find the angle between the lines whose direction cosines l, m, n satisfy the equations 5l + m + 3n = 0 and 5mn − 2nl + 6lm = 0.
If the vectors `3hat"i" + 5hat"k", 4hat"i" + 2hat"j" - 3hat"k"` and `3hat"i" + hat"j" + 4hat"k"` are the coterminus edges of the parallelopiped, then find the volume of the parallelopiped.
If the vectors `- 3hati + 4hatj - 2hatk , hati + 2hatk` and `hati - phatj` are coplanar, then find the value of p.
If `bar"u" = hat"i" - 2hat"j" + hat"k" , bar"v" = 3hat"i" + hat"k"` and `bar"w" = hat"j" - hat"k"` are given vectors, then find `(bar"u" + bar"w").[(bar"u" xx bar"v") xx (bar"v" xx bar"w")]`
If `vec"a", vec"b", vec"c"` are three non-coplanar vectors represented by concurrent edges of a parallelepiped of volume 4 cubic units, find the value of `(vec"a" + vec"b") * (vec"b" xx vec"c") + (vec"b" + vec"c")* (vec"c" xx vec"a") + (vec"c" + vec"a") * (vec"a" xx vec"b")`
If the vectors `"a"hat"i" + "a"hat"j" + "c"hat"k", hat"i" + hat"k"` and `"c"hat"i" + "c"hat"j" + "b"hat"k"` are coplanar, prove that c is the geometric mean of a and b
If the scalar triple product of the vectors `-3hat"i" + 7hat"j" - 3hat"k", 3hat"i" - 7hat"j" + lambdahat"k" and 7hat"i" - 5hat"j" - 5hat"j"` is 272 then λ = ______.
Let `bar"a", bar"b", bar"c"` be three vectors such that `bar"a" ≠ 0`, and `bar"a" xx bar"b" = 2bar"a" xx bar"c", |bar"a"| = |bar"c"| = 1, |bar"b"| = 4` and `|bar"b" xx bar"c"| = sqrt(15)`. If `bar"b" - 2bar"c" = lambdabar"a"`, then λ is equal to ______.
If θ is the angle between the unit vectors `bar"a"` and `bar"b"`, the `cos theta = theta/2` = ______.
If `veca = hati + hatj + hatk, veca.vecb` = 1 and `veca xx vecb = hatj - hatk`, then find `|vecb|`.
If `veca, vecb, vecc` are three non-coplanar vectors, then the value of `(veca.(vecb xx vecc))/((vecc xx veca).vecb) + (vecb.(veca xx vecc))/(vecc.(veca xx vecb))` is ______.
Prove that the volume of a tetrahedron with coterminus edges `overlinea, overlineb` and `overlinec` is `1/6[(overlinea, overlineb, overlinec)]`.
Hence, find the volume of tetrahedron whose coterminus edges are `overlinea = hati + 2hatj + 3hatk, overlineb = -hati + hatj + 2hatk` and `overlinec = 2hati + hatj + 4hatk`.
Find the volume of the parallelopiped whose vertices are A (3, 2, −1), B (−2, 2, −3) C (3, 5, −2) and D (−2, 5, 4).
If `u=hati -2hatj + hatk, barr=3hati + hatk and w=hatj, hatk` are given vectors, then find `[baru + barw]. [(barw xx barr)xx(barr xx barw)]`
