Advertisements
Advertisements
प्रश्न
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
Advertisements
उत्तर
Let `vec"a"_1 = "a"vec"i" + "a"vec"j" + "c"vec"k"`
`vec"a"_2 = vec"i" + vec"k"`
`vec"a"_3 = "c"vec"i" + "c"vec"j" + "b"vec"k"`
But `vec"a"_1, vec"a"_2, vec"a"_3` are coplanar .......(Given)
So, `[(vec"a"_1, vec"a"_2, vec"a"_3)]` = 0
`|("a", "a", "c"),(1, 0, 1),("c", "c", "b")|` = 0
⇒ `"a"[0 - "c"] - "a"["b" - "c"] + "c"["c" - 0]` = 0
⇒ `- "ac" - "Ab" + "ac" + "c"^2` = 0
⇒ c2 = ab
⇒ c = `sqrt("ab")`
∴ c is the geometric means of ‘a’ and ‘b’.
APPEARS IN
संबंधित प्रश्न
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 `
If `bar c = 3bara- 2bar b ` Prove that `[bar a bar b barc]=0`
Find λ, if the vectors `veca=hati+3hatj+hatk,vecb=2hati−hatj−hatk and vecc=λhatj+3hatk` 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 the parallelopiped whose coterminus edges are given by vectors `2hati+5hatj-4hatk, 5hati+7hatj+5hatk and 4hati+5hatj-2hatk`
Evaluate the following:
\[\left[\hat{i}\hat{j}\hat{k} \right] + \left[ \hat{j}\hat{k}\hat {i} \right] + \left[ \hat{k}\hat{i} \hat{j} \right]\]
Find the volume of the parallelopiped whose coterminous edges are represented by the vector:
\[\vec{a} = 2 \hat{i} + 3 \hat{j} + 4 \hat{k} , \vec{b} =\hat{ i} + 2 \hat{j} - \hat{k} , \vec{c} = 3 \hat{i} - \hat{j} + 2 \hat{k}\]
Prove that: \[\left( \vec{a} - \vec{b} \right) \cdot \left\{ \left( \vec{b} - \vec{c} \right) \times \left( \vec{c} - \vec{a} \right) \right\} = 0\]
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.
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
\[\left( \vec{a} + \vec{b} \right) \cdot \left( \vec{b} + \vec{c} \right) \times \left( \vec{a} + \vec{b} + \vec{c} \right) =\]
Determine where `bb(bara)` and `bb(barb)` are orthogonal, parallel or neither.
`bara = -3/5hati + 1/2hatj + 1/3hatk , barb = 5hati + 4hatj + 3hatk`
Let `vec"a", vec"b", vec"c"` be three non-zero vectors such that `vec"c"` is a unit vector perpendicular to both `vec"a"` and `vec"b"`. If the angle between `vec"a"` and `vec"b"` is `pi/6`, show that `[(vec"a", vec"b", vec"c")]^2 = 1/4|vec"a"|^2|vec"b"|^2`
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 `veca = hati + hatj + hatk, veca.vecb` = 1 and `veca xx vecb = hatj - hatk`, then find `|vecb|`.
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 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.
If `barc = 3bara - 2barb` and `[bara barb + barc bara + barb + barc]` = 0 then prove that `[bara barb barc]` = 0
Determine whether `bb(bara and barb)` are orthogonal, parallel or neither.
`bara=-3/5hati+1/2hatj+1/3hatk,barb=5hati+4hatj+3hatk`
