Advertisements
Advertisements
Question
Find the volume of the parallelopiped whose coterminous edges are represented by the vector:
\[\vec{a} = \hat{i} + \hat{j} + \hat{k} , \vec{b} =\hat{ i} - \hat{j} + \hat{k} , \vec{c} = \hat{i} + 2 \hat{j} - \hat{k}\]
Solution
Given:
\[ \vec{a} = \hat{i} + \hat{j} + \hat{k} \]
\[ \vec{b} = \hat{i} -\hat{ j} + k \]
\[ \vec{c} =\hat{ i} + 2j - \hat{k} \]
\[\text{We know that the volume of a parallelopiped whose three adjacent edges are } \vec{a} , \vec{b} , \vec{c}\text{ is equal to } \left| \left[ \vec{a} \vec{b} \vec{c} \right] \right| . \]
Here,
\[\left[ \vec{a} \vec{b} \vec{c} \right] = \begin{vmatrix}1 & 1 & 1 \\ 1 & - 1 & 1 \\ 1 & 2 & - 1\end{vmatrix} = 1\left( 1 - 2 \right) - 1\left( - 1 - 1 \right) + 1\left( 2 + 1 \right) = 4\]
\[\text { Volume of the parallelopiped }= \left| \left[ \vec{a} \vec{b} \vec{c} \right] \right| = \left| 4 \right| = 4 \text { cubic units }\]
APPEARS IN
RELATED QUESTIONS
Show that the four points A(4, 5, 1), B(0, –1, –1), C(3, 9, 4) and D(–4, 4, 4) are coplanar.
Find λ, if the vectors `veca=hati+3hatj+hatk,vecb=2hati−hatj−hatk and vecc=λhatj+3hatk` are coplanar.
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
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 \[\left[ \vec{a} \vec{b} \vec{c} \right]\] , 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:
\[\vec{a} = 11 \hat{i} , \vec{b} = 2 \hat{j} , \vec{c} = 13 \hat{k}\]
Find the value of λ so that the following vector is coplanar:
\[\vec{a} = \hat{i} - \hat{j} + \hat{k} , \vec{b} = 2 \hat {i} + \hat {j} - \hat {k} , \vec{c} = \lambda\hat { i} - \hat {j} + \lambda \hat {k}\]
Find the value of λ so that the following vector is coplanar:
\[\vec{a} = 2 \hat{i} - \hat {j} + \hat {k} , \vec{b} = \hat {i} + 2 \hat {j} - 3 \hat {k} , \vec{c} = \lambda \hat {i} + \lambda \hat {j} + 5 \hat {k}\]
Show that the points A (−1, 4, −3), B (3, 2, −5), C (−3, 8, −5) and D (−3, 2, 1) are coplanar.
Find the values of 'a' for which the vectors
\[\vec{\alpha} = \hat {i} + 2 \hat {j} + \hat {k} , \vec{\beta} = a \hat {i} + \hat {j} + 2 \hat {k} \text { and } \vec{\gamma} = \hat {i} + 2 \hat {j} + a \hat { k }\] are coplanar.
Find \[\vec{a} . \left( \vec{b} \times \vec{c} \right)\], if \[\vec{a} = 2 \hat {i} + \hat {j} + 3 \hat {k} , \vec{b} = - \hat {i} + 2 \hat {j} + \hat {k}\] and \[\vec{c} = 3 \hat { i} + \hat {j} + 2 \hat {k}\].
If \[\vec{a,} \vec{b,} \vec{c}\] are three non-coplanar mutually perpendicular unit vectors, then \[\left[ \vec{a} \vec{b} \vec{c} \right],\] 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
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 =
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
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.
Determine where `bar"a"` and `bar"b"` are orthogonal, parallel or neithe:
`bar"a" = - 9hat"i" + 6hat"j" + 15hat"k"` , `bar"b" = 6hat"i" - 4hat"j" - 10hat"k"`.
Prove by vector method, that the angle subtended on semicircle is a right angle.
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 `- 3hat"i" + 4hat"j" - 2hat"k" , hat"i" + 2hat"k"` and `hat"i" - "p"hat"j"` 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")]`
Using properties of scalar triple product, prove that `[(bar"a" + bar"b", bar"b" + bar"c", bar"c" + bar"a")] = 2[(bar"a", bar"b", bar"c")]`.
If `vec"a" = hat"i" - hat"k", vec"b" = xhat"i" + hat"j" + (1 - x)hat"k", vec"c" = yhat"i" + xhat"j" + (1 + x - y)hat"k"`, show that `[(vec"a", vec"b", vec"c")]` depends on neither x nor y
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 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
If the points having the position vectors `2hat"i" + hat"j" - hat"k", -hat"j", 4hat"i" + 4hat"k"` and `lambdahat"i" + hat"k"` lie on the same plane, then λ is equal to ______.
Determine whether `bara and barb` are orthogonal, parallel or neither.
`bara = - 3/5 hati+ 1/2 hatj + 1/3 hatk , barb= 5hati + 4hatj + 3hatk`
Determine whether `bara` and `barb` are orthogonal, parallel or neither.
`bara = - 3/5 hati + 1/2 hatj + 1/3 hatk, barb = 5hati + 4hatj + 3hatk`
Determine whether `bb(bara and barb)` are orthogonal, parallel or neither.
`bar a = -3/5hati + 1/2hatj + 1/3hatk, barb = 5hati + 4hatj + 3hatk`
Find the volume of the parallelopiped whose coterminous edges are `2hati - 3hatj, hati + hatj - hatk` and `3hati - hatk`.
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" xx bar"v" bar"u" xx bar"w" bar"v" xx bar"w"]`
If `barc = 3bara - 2barb` and `[bara barb + barc bara + barb + barc]` = 0 then prove that `[bara barb barc]` = 0
If `baru = hati - 2hatj + hatk, barv = 3hati + hatk "and" barw = hatj - hatk` are given vectors, then find `[baru + barw]·[(baru xx barv)xx(barv xx barw)]`
