Advertisements
Advertisements
Question
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.
Advertisements
Solution
\[\text {Let:} \vec{a} = \left( \sec^2 A \right) \hat {i} + \hat {j} + \hat {k} , \vec{b} = \hat {i} + \left( \sec^2 B \right) \hat {j} + \hat {k} \text{and} \vec{c} = \hat {i} + \hat {j} + \left( \sec^2 C \right) \hat {k} \]
\[\text { We know that three vectors are coplanar iff their scaler triple product is zero . i . e .} , \left[ \vec{a} \vec{b} \vec{c} \right] = 0\]
\[\text { Here,} \left[ \vec{a} \vec{b} \vec{c} \right] = 0\]
\[ \Rightarrow \begin{vmatrix}\sec^2 A & 1 & 1 \\ 1 & \sec^2 B & 1 \\ 1 & 1 & \sec^2 C\end{vmatrix} = 0 \]
\[ \Rightarrow \sec^2 A\left[ \left( \sec^2 B \times \sec^2 C \right) - 1 \right] - 1\left( \sec^2 C - 1 \right) + 1\left( 1 - \sec^2 B \right) = 0\]
\[ \Rightarrow \sec^2 A \sec^2 B \sec^2 C - \sec^2 A - \sec^2 C + 1 + 1 - \sec^2 B = 0\]
\[ \Rightarrow \left( 1 + \tan^2 A \right)\left( 1 + \tan^2 B \right) \left( 1 + \tan^2 C \right) - \left( 1 + \tan^2 A \right) - \left( 1 + \tan^2 C \right) + 1 + 1 - \left( 1 + \tan^2 B \right) = 0\]
\[\Rightarrow 1 + \tan^2 A + \tan^2 B + \tan^2 C + \tan^2 A \tan^2 B + \tan^2 B \tan^2 C + \tan^2 C \tan^2 A + \tan^2 A \tan^2 B \tan^2 C - 1 - \tan^2 A - 1 - \tan^2 C + 1 + 1 - 1 - \tan^2 B = 0\]
\[ \Rightarrow \tan^2 A \tan^2 B + \tan^2 B \tan^2 C + \tan^2 C \tan^2 A + \tan^2 A \tan^2 B \tan^2 C = 0\]
\[ \Rightarrow \tan^2 A \tan^2 B + \tan^2 B \tan^2 C + \tan^2 C \tan^2 A = - \tan^2 A \tan^2 B \tan^2 C\]
\[ \Rightarrow \frac{\tan^2 A \tan^2 B + \tan^2 B \tan^2 C + \tan^2 C \tan^2 A}{\tan^2 A \tan^2 B \tan^2 C} = - 1\]
\[ \Rightarrow \cot^2 C + \cot^2 A + \cot^2 B = - 1\]
\[ \Rightarrow {cosec}^2 C - 1 + {cosec}^2 A - 1 + {cosec}^2 B - 1 = - 1\]
\[ \therefore {cosec}^2 A + {cosec}^2 B + {cosec}^2 C = 2\]
APPEARS IN
RELATED QUESTIONS
Prove that the volume of a parallelopiped with coterminal edges as ` bara ,bar b , barc `
Hence find the volume of the parallelopiped with coterminal edges `bar i+barj, barj+bark `
Prove that, for any three vector `veca,vecb,vecc [vec a+vec b,vec b+vec c,vecc+veca]=2[veca vecb vecc]`
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).
Find the value of λ, if four points with position vectors `3hati + 6hatj+9hatk`, `hati + 2hatj + 3hatk`,`2hati + 3hatj + hatk` and `4hati + 6hatj + lambdahatk` are coplanar.
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
Show that the four points A, B, C and D with position vectors `4hati + 5hatj + hatk`, `-hatj-hatk`, `3hati + 9hatj + 4hatk` and `4(-hati + hatj + hatk)` respectively are coplanar
Find the volume of a parallelopiped whose edges are represented by the vectors:
`vec a = 2 hat i - 3 hat j - 4 hat k`, `vec b = hat i + 2 hat j - hat k` and `vec c = 3 hat i + hat j + 2 hatk`
Find the volume of the parallelopiped whose coterminus edges are given by vectors `2hati+5hatj-4hatk, 5hati+7hatj+5hatk and 4hati+5hatj-2hatk`
Find the volume of the parallelopiped whose coterminous edges are represented by the vector:
\[\vec{a} = 2 \hat{i} - 3 \hat{j} + 4 \hat{k} , \vec{b} = \hat{i} + 2 \hat{j} - \hat{k} , \vec{c} = 3 \hat{i} - \hat{j} - 2 \hat{k}\]
Find the volume of the parallelopiped whose coterminous edges are represented by the vector:
\[\vec{a} = 11 \hat{i} , \vec{b} = 2 \hat{j} , \vec{c} = 13 \hat{k}\]
Show of the following triad of vector is coplanar:
\[\vec{a} = \hat {i} + 2 \hat{j} - \hat {k} , \vec{b} = 3 \hat {i} + 2 \hat{j} + 7 \hat {k} , \vec{c} = 5 \hat {i} + 6 \hat { j} + 5 \hat {k}\]
Show of the following triad of vector is coplanar:
\[\vec{a} = - 4 \hat{i} - 6 \hat{j} - 2 \hat{k} , \vec{b} = -\hat{ i} + 4 \hat{j} + 3 \hat{k} , \vec{c} = - 8 \hat{i} - \hat{j} + 3 \hat{k}\]
Show of the following triad of vector is coplanar:
\[\hat{a} = \hat{i} - 2 \hat {j} + 3 \hat {k} , \hat {b} = - 2 \hat {i} + 3 \hat {j} - 4 \hat { k}, \hat {c} = \hat { i} - 3 \hat { j} + 5 \hat { k }\]
Show that the points A (−1, 4, −3), B (3, 2, −5), C (−3, 8, −5) and D (−3, 2, 1) are coplanar.
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.
If \[\vec{r} \cdot \vec{a} = \vec{r} \cdot \vec{b} = \vec{r} \cdot \vec{c} = 0\] for some non-zero vector \[\vec{r} ,\] then the value of \[\left[ \vec{a} \vec{b} \vec{c} \right],\] is
If \[\vec{a,} \vec{b,} \vec{c}\] are non-coplanar vectors, then \[\frac{\vec{a} \cdot \left( \vec{b} \times \vec{c} \right)}{\left( \vec{c} \times \vec{a} \right) \cdot \vec{b}} + \frac{\vec{b} \cdot \left( \vec{a} \times \vec{c} \right)}{\vec{c} \cdot \left( \vec{a} \times \vec{b} \right)}\] is equal to
\[\left[ \vec{a} \vec{b} \vec{a} \times \vec{b} \right] + \left( \vec{a} . \vec{b} \right)^2 =\]
\[\left( \vec{a} + \vec{b} \right) \cdot \left( \vec{b} + \vec{c} \right) \times \left( \vec{a} + \vec{b} + \vec{c} \right) =\]
If \[\vec{a,} \vec{b,} \vec{c}\] are three non-coplanar vectors, then \[\left( \vec{a} + \vec{b} + \vec{c} \right) . \left[ \left( \vec{a} + \vec{b} \right) \times \left( \vec{a} + \vec{c} \right) \right]\] equals
\[\left( \vec{a} + 2 \vec{b} - \vec{c} \right) \cdot \left\{ \left( \vec{a} - \vec{b} \right) \times \left( \vec{a} - \vec{b} - \vec{c} \right) \right\}\] is equal to
Find the volume of the parallelopiped, if the coterminus edges are given by the vectors `2hat"i" + 5hat"j" -4 hat"k", 5hat"i" +7hat"j"+5 hat "k" , 4hat"i" +5hat"j" - 2 hat"k"`.
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.
If a line has the direction ratios 4, −12, 18, then find its direction cosines
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.
Find the volume of the parallelepiped whose coterminous edges are represented by the vectors `- 6hat"i" + 14hat"j" + 10hat"k", 14hat"i" - 10hat"j" - 6hat"k"` and `2hat"i" + 4hat"j" - 2hat"k"`
Find the altitude of a parallelepiped determined by the vectors `vec"a" = - 2hat"i" + 5hat"j" + 3hat"k", vec"b" = hat"i" + 3hat"j" - 2hat"k"` and `vec"c" = - vec"i" + vec"j" + 4vec"k"` if the base is taken as the parallelogram determined by `vec"b"` and `vec"c"`
The volume of tetrahedron whose vertices are A(3, 7, 4), B(5, -2, 3), C(-4, 5, 6), D(1, 2, 3) is ______.
If the volume of the tetrahedron formed by the coterminous edges `bar"a", bar"b" and bar"c"` is 5, then the volume of the parallelopiped formed by the coterminous edges `bar"a" xx bar"b", bar"b" xx bar"c" and bar"c" xx bar"a"` is
Determine whether `bara` and `barb` are orthogonal, parallel or neither.
`bara = - 3/5 hati + 1/2 hatj + 1/3 hatk, barb = 5hati + 4hatj + 3hatk`
Find the volume of the parallelopiped whose coterminous edges are `2hati - 3hatj, hati + hatj - hatk` and `3hati - hatk`.
If the points A(1, 2, 3), B(–1, 1, 2), C(2, 3, 4) and D(–1, x, 0) are coplanar find the value of x.
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 `barc = 3bara - 2barb` and `[bara barb + barc bara + barb + barc]` = 0 then prove that `[bara barb barc]` = 0
If a vector has direction angles 45ºand 60º find the third direction angle.
Determine whether `\bb(bara and barb)` are orthogonal, parallel or neither.
`bara = -3/5 hati + 1/2 hatj + 1/3 hatk, barb = 5hati + 4hatj + 3hatk `
