Advertisements
Advertisements
Question
The volume of the parallelepiped whose coterminus edges are `7hat"i" + lambdahat"j" - 3hat"k", hat"i" + 2hat"j" - hat"k", -3hat"i" + 7hat"j" + 5hat"k"` is 90 cubic units. Find the value of λ
Advertisements
Solution
Volume of the parallelepiped = `[bar"a", bar"b", bar"c"]`
`|(7, lambda, -3),(1, 2, -1),(-3, 7, 5)|` = 90
7(10 + 7) – λ(5 – 3) – 3(7 + 6) = 90
7(17) – λ(2) – 3(13) = 90
119 – 2λ – 39 = 90
2λ = 119 – 39 – 90
2λ = – 10
λ = – 5
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.
If `bara=3hati-hatj+4hatk, barb=2hati+3hatj-hatk, barc=-5hati+2hatj+3hatk` then `bara.(barbxxbarc)=`
(A) 100
(B) 101
(C) 110
(D) 109
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 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 \[\left[ 2 \hat { i } \ 3 \hat { j }\ 4 \hat { k } \right] .\]
If \[\vec{a,} \vec{b}\] \[\text { are non-collinear vectors, then find the value of} \left[ \vec{a} \vec{b}\hat { i} \right] \hat{i} + \left[ \vec{a} \vec{b} \hat {j} \right] \hat {j} + \left[ \vec{a} \vec{b} \hat {k} \right] \hat {k} .\]
For any three vectors \[\vec{a,} \vec{b,} \vec{c}\] the expression \[\left( \vec{a} - \vec{b} \right) . \left\{ \left( \vec{b} - \vec{c} \right) \times \left( \vec{c} - \vec{a} \right) \right\}\] equals
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
\[\left[ \vec{a} \vec{b} \vec{a} \times \vec{b} \right] + \left( \vec{a} . \vec{b} \right)^2 =\]
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
\[\left( \vec{a} + 2 \vec{b} - \vec{c} \right) \cdot \left\{ \left( \vec{a} - \vec{b} \right) \times \left( \vec{a} - \vec{b} - \vec{c} \right) \right\}\] is equal to
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"`.
Determine where `bar"a"` and `bar"b"` are orthogonal, parallel or neithe:
`bar"a" = 4hat"i" - hat"j" + 6hat"k"` , `bar"b" = 5hat"i" - 2hat"j" + 4hat"k"`
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 `vec"a" = hat"i" - 2hat"j" + 3hat"k", vec"b" = 2hat"i" + hat"j" - 2hat"k", vec"c" = 3hat"i" + 2hat"j" + hat"k"`, find `vec"a" * (vec"b" xx vec"c")`
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 `bb(bara and barb)` are orthogonal, parallel or neither.
`bar a = -3/5hati + 1/2hatj + 1/3hatk, barb = 5hati + 4hatj + 3hatk`
Determine whether `bb(bara and barb)` are orthogonal, parallel or neither.
`bara=-3/5hati+1/2hatj+1/3hatk,barb=5hati+4hatj+3hatk`
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 volume of a tetrahedron whose vertices are A(−1, 2, 3) B(3, −2, 1), C(2, 1, 3) and D(−1, −2, 4).
