Advertisements
Advertisements
प्रश्न
If `veca = hati + hatj + hatk, veca.vecb` = 1 and `veca xx vecb = hatj - hatk`, then find `|vecb|`.
Advertisements
उत्तर
Given, `veca = hati + hatj + hatk`
`veca.vecb` = 1
And `veca xx vecb = hatj - hatk`
Let `vecb = ahati + bhatj + chatk`
Now, `veca.vecb` = 1
⇒ `(hati + hatj + hatk)(b_1hati + b_2hatj + b_3hatk)` = 1
⇒ `b_1 + b_2 + b_3` = 1 ...(i)
And `veca xx vecb = hatj - hatk`
⇒ `|(hati, hatj, hatk),(1, 1, 1),(b_1, b_2, b_3)| = hatj - hatk`
⇒ `hati(b_3 - b_2) - hatj(b_3 - b_1) + hatk(b_2 - b_1) = hatj - hatk`
On comparing both sides, we get
–(b3 – b1) = 1 and b2 – b1 = –1
⇒ b3 – b1 = –1 and b2 – b1 = –1
⇒ b3 = –1 + a and b2 = –1 + b1 ...(ii)
Now from equation (i), we get
b1 + (–1 + b1) + (–1 + b1) = 1
⇒ 3b1 = 3
⇒ b1 = 1
From equation (ii), we get
b2 = 0 and b3 = 0
∴ `vecb = hati`
Therefore, `|vecb|` = 1
APPEARS IN
संबंधित प्रश्न
Prove that, for any three vector `veca,vecb,vecc [vec a+vec b,vec b+vec c,vecc+veca]=2[veca vecb vecc]`
If `bara=3hati-hatj+4hatk, barb=2hati+3hatj-hatk, barc=-5hati+2hatj+3hatk` then `bara.(barbxxbarc)=`
(A) 100
(B) 101
(C) 110
(D) 109
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
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
Prove that a necessary and sufficient condition for three vectors \[\vec{a}\], \[\vec{b}\], \[\vec{c}\] to be coplanar is that there exist scalars l, m, n not all zero simultaneously such that \[l \vec{a} + m \vec{b} + n \vec{c} = \vec{0} .\]
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 \[\vec{a} = 2 \hat{i} - 3 \hat{j} , \vec{b} = \hat{i} + \hat{j} - \hat{k} \text{ and } \vec{c} = 3 \hat{i} - \hat{k}\]
Show of the following triad of vector is coplanar:
\[\vec{a} = \hat {i} + 2 \hat{j} - \hat {k} , \vec{b} = 3 \hat {i} + 2 \hat{j} + 7 \hat {k} , \vec{c} = 5 \hat {i} + 6 \hat { j} + 5 \hat {k}\]
Find the value of λ so that the following vector is coplanar:
\[\vec{a} = \hat{i} - \hat{j} + \hat{k} , \vec{b} = 2 \hat {i} + \hat {j} - \hat {k} , \vec{c} = \lambda\hat { i} - \hat {j} + \lambda \hat {k}\]
Show that the points A (−1, 4, −3), B (3, 2, −5), C (−3, 8, −5) and D (−3, 2, 1) are coplanar.
Show that four points whose position vectors are
\[6 \hat { i} - 7 \hat {j} , 16 \hat { i} - 19 \hat { j} - 4 \hat {k} , 3 \hat {i} - 6 \hat {k} , 2 \hat { i} - 5 \hat {j}+ 10 \hat {k}\]
Prove that: \[\left( \vec{a} - \vec{b} \right) \cdot \left\{ \left( \vec{b} - \vec{c} \right) \times \left( \vec{c} - \vec{a} \right) \right\} = 0\]
If four points A, B, C and D with position vectors 4 \[\hat { i} +3\] \[\hat { j} +3\] \[\hat { k} ,5\] \[\hat { i} +\] \[x\hat { j} +7\] \[\hat { k} ,5\] \[\hat { i} +3\] \[\hat { j}\] and \[7 \hat{i} + 6 \hat{j} + \hat{k}\] respectively are coplanar, then find the value of x.
Write the value of \[\left[ 2 \hat { i } \ 3 \hat { j }\ 4 \hat { k } \right] .\]
If \[\vec{a,} \vec{b,} \vec{c}\] are non-coplanar vectors, then find the value of \[\frac{\vec{a} \cdot \left( \vec{b} \times \vec{c} \right)}{\left( \vec{c} \times \vec{a} \right) \cdot \vec{b}} + \frac{\vec{b} \cdot \left( \vec{a} \times \vec{c} \right)}{\vec{c} \cdot \left( \vec{a} \times \vec{b} \right)}\].
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
If a line has the direction ratios 4, −12, 18, then find its direction cosines
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 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"`
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 ______.
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 vertices are A (3, 2, −1), B (−2, 2, −3) C (3, 5, −2) and D (−2, 5, 4).
If a vector has direction angles 45ºand 60º find the third direction angle.
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).
The magnitude of a vector which is orthogonal to the vector \[\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}\] and is coplanar with the vectors \[\hat{\mathrm{i}}+\hat{\mathrm{j}}+2\hat{\mathrm{k}}\] and \[\hat{\mathrm{i}}+2\hat{\mathrm{j}}+\hat{\mathrm{k}}\] is
