Recent questions tagged ch2

Prove that $( \overrightarrow{a}\times \overrightarrow{b})\times \overrightarrow{c}= \overrightarrow{a}\times (\overrightarrow{b}\times \overrightarrow{c})$ if $ \overrightarrow{a}$ and $ \overrightarrow{c}$ are collinear. (Where vector triple product is non-zero).

asked Apr 5, 2013 by poojasapani_1

1 answer

If $ \overrightarrow{a}= \overrightarrow{2i}+ \overrightarrow{3j}- \overrightarrow{5k}, \overrightarrow{b}= -\overrightarrow{1}+ \overrightarrow{j}+ \overrightarrow{2k}$ and $ \overrightarrow{c}= \overrightarrow{4i}- \overrightarrow{2j}+ \overrightarrow{3k}, $ Show that $( \overrightarrow{a}\times \overrightarrow{b})\times \overrightarrow{c} \neq \overrightarrow{a} \times( \overrightarrow{b}\times \overrightarrow{c})$

asked Apr 5, 2013 by poojasapani_1

1 answer

Prove thet $ \overrightarrow{a}\times( \overrightarrow{b}\times \overrightarrow{c})+ \overrightarrow{b}\times( \overrightarrow{c}\times{a})+ \overrightarrow{c}\times( \overrightarrow{a}\times \overrightarrow{b})=0$

asked Apr 5, 2013 by poojasapani_1

1 answer

if $\ \overrightarrow{a}= \overrightarrow{2i}+ \overrightarrow{3j}- \overrightarrow{k} , \overrightarrow{b}= -\overrightarrow{2i}+ \overrightarrow{5k}, \overrightarrow{c}= \overrightarrow{j}- \overrightarrow{3k}. $ Verify that $ \overrightarrow{a}\times ( \overrightarrow{b}\times \overrightarrow{c})= (\overrightarrow{a}. \overrightarrow{c}) \overrightarrow{b}- (\overrightarrow{a}. \overrightarrow{b})c$

asked Apr 5, 2013 by poojasapani_1

1 answer

Show that the points$(1 ,3 ,1), (1, 1, -1),(-1,1, 1),(2 ,2,- 1) $ are lying on the same plane.(Hint : It is enough to prove any three vectors formed by these four points are coplanar).

asked Apr 5, 2013 by poojasapani_1

1 answer

Prove that $\mid[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}]\mid=a b c $ if and only if $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are mutually perpendicular.

asked Apr 5, 2013 by poojasapani_1

1 answer

The volume of a parallelopiped whose edges are represented by $-\overrightarrow{12i}+\lambda\overrightarrow{k}, \overrightarrow{3j}-\overrightarrow{k}, \overrightarrow{2i}+\overrightarrow{j}-\overrightarrow{15k} $ is $546;\quad$ find the value of $\lambda$.

asked Apr 5, 2013 by poojasapani_1

1 answer

Show that the vectors $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are coplanar if and only if $\overrightarrow{a}+\overrightarrow{b},\overrightarrow{b}+\overrightarrow{c},\overrightarrow{c}+\overrightarrow{a}$ are coplanar.

asked Apr 5, 2013 by poojasapani_1

1 answer

Find the magnitude and direction cosines of the moment about the point $(1, -2 ,3)$ of a force $\overrightarrow{2i}+\overrightarrow{3j}+\overrightarrow{6k}$ Whose line of action passes through the origin.

asked Apr 5, 2013 by poojasapani_1

1 answer

Show that the torque about the point $ A(3, -1, 3 )$ of a force $\overrightarrow{4i}+\overrightarrow{2j}\overrightarrow{k}$ throught the point $\overrightarrow\;B(5, 2, 4)$ is $\overrightarrow{i}+\overrightarrow{2j}-\overrightarrow{8k}$.

asked Apr 5, 2013 by poojasapani_1

1 answer

Forces $\overrightarrow{2i}+\overrightarrow{7j}, \overrightarrow{2i}-\overrightarrow{5j}+\overrightarrow{6k}, \overrightarrow{-i}+\overrightarrow{2j}-\overrightarrow{k}$ act at a point $P$ Whose position vector is $\overrightarrow{4i}-\overrightarrow{3j}-\overrightarrow{2k}.$ Find the moment of the resultant of three forces acting at $P$ about the point $Q$ whose position vector is $\overrightarrow{6i}+\overrightarrow{j}-\overrightarrow{3k}.$

asked Apr 5, 2013 by poojasapani_1

1 answer

Prove that sin $(A - B)$= sin $ A$ cos $B$ - cos $A$ sin $B$.

asked Apr 5, 2013 by poojasapani_1

1 answer

Prove that twice the area of parallelogram is equal to the area of another parallelogram formed by taking as its adjacent sides the diagonals of the former parallelogram.

asked Apr 5, 2013 by poojasapani_1

1 answer

Prove by the vector method, thet the parallelogram on the same base and between the same parallels are equal in area.

asked Apr 5, 2013 by poojasapani_1

1 answer

Find the area of the triangle whose vertices are $(3, -1, 2), (1 ,-1, -3 ), $and $(4, -3, 1) $

asked Apr 5, 2013 by poojasapani_1

1 answer

Find the area of the parallelogram determined by the sides $\overrightarrow{i}+\overrightarrow{2j}+\overrightarrow{3k}$ and $-\overrightarrow{3i}-\overrightarrow{2j}+\overrightarrow{k}$

asked Apr 5, 2013 by poojasapani_1

1 answer

Find the area of the parallelogram whose diagonals are represented by $\overrightarrow{2i}+\overrightarrow{3j}+\overrightarrow{6k}$ and $\overrightarrow{3i}-\overrightarrow{6j}+\overrightarrow{2k}$

asked Apr 5, 2013 by poojasapani_1

1 answer

Find the area of parallelogram $A\;B\;C\;D$ whose vertices are $A(-5 ,2 ,5),B(-3 ,6 ,7),C(4 ,-1 ,5)$ and $D(2 ,-5 ,3)$

asked Apr 5, 2013 by poojasapani_1

1 answer

If $\overrightarrow{a}\times \overrightarrow{b}= \overrightarrow{C}\times \overrightarrow{d}$ and $\overrightarrow{a}\times \overrightarrow{c}=\overrightarrow{b}\times\overrightarrow{d},$ show that $\overrightarrow{a}- \overrightarrow{d}$ and $\overrightarrow{b}-\overrightarrow{c}$ are parallel.

asked Apr 4, 2013 by poojasapani_1

1 answer

Let $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ be unit vectors such that $\overrightarrow{a}.\overrightarrow{b}=\overrightarrow{a}.\overrightarrow{c}=0$ and the angle between $\overrightarrow{b}$ and $\overrightarrow{c}$ is $\Large\frac{\pi}{6}.$ Prove that $\overrightarrow{a}=\pm 2(\overrightarrow{b}\times\overrightarrow{c})$

asked Apr 4, 2013 by poojasapani_1

1 answer

For any three vectors $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ show that $\overrightarrow{a}\times(\overrightarrow{b}+\overrightarrow{c})+\overrightarrow{b}\times(\overrightarrow{c}+\overrightarrow{a})+\overrightarrow{c}\times(\overrightarrow{a}+\overrightarrow{b})=\overrightarrow{0}$

asked Apr 4, 2013 by poojasapani_1

1 answer

If $\overrightarrow{a}=\overrightarrow{i}+\overrightarrow{3j}-\overrightarrow{2k}$ and $\overrightarrow{b}=\overrightarrow{-i}+\overrightarrow{3k}$ than find $\overrightarrow{a}\times\overrightarrow{b}$. Verify that $\overrightarrow{a}$ and $\overrightarrow{b}$ are perpendicular to$\overrightarrow{a}\times\overrightarrow{b}$ separately.

asked Apr 4, 2013 by poojasapani_1

1 answer

If $|\overrightarrow{a}|=2, |\overrightarrow{b}|=7$ and $\overrightarrow{a}\times\overrightarrow{b}=\overrightarrow{3i}-\overrightarrow{2j}+\overrightarrow{6k}$ find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$

asked Apr 4, 2013 by poojasapani_1

1 answer

Find the angle between two vectors $\overrightarrow{a} $ and $\overrightarrow{b} $ if $|\overrightarrow{a} \times \overrightarrow{b}|=\overrightarrow{a}.\overrightarrow{b}$

asked Apr 4, 2013 by poojasapani_1

1 answer

Find the vectors whose length $5$ and which are perpendicular to the vectors $\overrightarrow{a}=\overrightarrow{3i}+\overrightarrow{j}-\overrightarrow{4k}$ and $\overrightarrow{b}=\overrightarrow{6i}+\overrightarrow{5j}-\overrightarrow{2k}$

asked Apr 4, 2013 by poojasapani_1

1 answer

Find the unit vectors perpendicular to the plane containing the vectors $\overrightarrow{2i}+\overrightarrow{j}+\overrightarrow{k}$ and $\overrightarrow{i}+\overrightarrow{2j}+\overrightarrow{k}$

asked Apr 4, 2013 by poojasapani_1

1 answer

If $|\overrightarrow{a}|=3, |\overrightarrow{b}|=4$ and $\overrightarrow{a}.\overrightarrow{b}=9$ than find $|\overrightarrow{a} \times \overrightarrow{b}|$

asked Apr 4, 2013 by poojasapani_1

1 answer

Find the magnitude of $\overrightarrow{a} \times \overrightarrow{b}$ if $\overrightarrow{a}=\overrightarrow{2i}+\overrightarrow{k}, \overrightarrow{b}=\overrightarrow{i}+\overrightarrow{j}+\overrightarrow{k}$

asked Apr 4, 2013 by poojasapani_1

1 answer

Forces of magnitudes $3$ and $4$ units acting in the directions $\overrightarrow{6i}+ \overrightarrow{2j} +\overrightarrow{3k}$ and $\overrightarrow{3i}-\overrightarrow{2j}+\overrightarrow{6k}$ respectively act on a particle which is displaced from the point $(2,2,-1)$to$( 4,3,1)$. find the work done by the forces.

asked Apr 4, 2013 by poojasapani_1

1 answer

The constant forces $\overrightarrow{2i}-\overrightarrow{5j}+ \overrightarrow{6k}, -\overrightarrow{i}+\overrightarrow{2j}-\overrightarrow{k},$and $\overrightarrow{2i}+\overrightarrow{7j} $ act on a particle which is displaced from position $\overrightarrow{4i}-\overrightarrow{3j}-\overrightarrow{2k}$ to position $\overrightarrow{6i}+\overrightarrow{j}-\overrightarrow{3k}$. find the work done.

asked Apr 4, 2013 by poojasapani_1

1 answer

A force magnitude $5$ units acting parallel to $\overrightarrow{2i}-\overrightarrow{2j}+ \overrightarrow{k}$ displaces the point of application from $(1,2,3)$to $(5,3,7) $ find the work done.

asked Apr 4, 2013 by poojasapani_1

1 answer

Find the work done by the force $f=\overrightarrow{2i} + \overrightarrow{j}+\overrightarrow{k}$ acting on particle, if the particle is displaced from the point with position vector $\overrightarrow{2i}+ \overrightarrow{2j}+ \overrightarrow{2k}$ to the point with the position vector $\overrightarrow{3i}+\overrightarrow{4j}+\overrightarrow{5k}$ .

asked Apr 4, 2013 by poojasapani_1

1 answer

Prove by the vector method , cos$(A+B)=$ cos$A$ cos$B$ - sin$A$ sin$B$

asked Apr 3, 2013 by poojasapani_1

1 answer

Prove by vector method , The sum of the squares of the diagonals of a parallelogram is equal to the sum of the squares of the sides.

asked Apr 3, 2013 by poojasapani_1

1 answer

Prove by vector method, The mid point of the hypotenuse of a right angled tringle is equidistant from its vertices.

asked Apr 3, 2013 by poojasapani_1

1 answer

Prove by the vector method If the diagonals of a parallelogram are equal than it is a rectangle.

asked Apr 3, 2013 by poojasapani_1

1 answer

Find the projection of $\overrightarrow{3i}+\overrightarrow{j}-\overrightarrow{k}$ on $\overrightarrow{4i}-\overrightarrow{j}+\overrightarrow{2k}$

asked Apr 3, 2013 by poojasapani_1

1 answer

Find the projection of $\overrightarrow{i}+\overrightarrow{2j}-\overrightarrow{2k}$ on $\overrightarrow{2i}-\overrightarrow{j}+ \overrightarrow{5k}$

asked Apr 3, 2013 by poojasapani_1

1 answer

Find the projection of $\overrightarrow{i}-\overrightarrow{j}$ on $z$ axis

asked Apr 3, 2013 by poojasapani_1

1 answer

Show that the points whose position vectors $\overrightarrow{4i}-\overrightarrow{3j}+\overrightarrow{k}, \overrightarrow{2i}-\overrightarrow{4j}+\overrightarrow{5k} ,\overrightarrow{i}-\overrightarrow{j}$ form a right angled tringle.

asked Apr 3, 2013 by poojasapani_1

1 answer

Show that the vectors $\overrightarrow{3i}-\overrightarrow{2j}+\overrightarrow{k},\overrightarrow{i}-\overrightarrow{3j}+\overrightarrow{5k}$ and $\overrightarrow{2i}+\overrightarrow{j}-\overrightarrow{4k}$ form a right angled tringle.

asked Apr 3, 2013 by poojasapani_1

1 answer

Let $\overrightarrow{u},\overrightarrow{v}$ and $\overrightarrow{w}$ be vector such that $\overrightarrow{u}+\overrightarrow{v}+\overrightarrow{w}=\overrightarrow{0}.$ If $|\overrightarrow{u}|=3, |\overrightarrow{v}|=4$ and $|\overrightarrow{w}|=5$ than find $\overrightarrow{u}.\overrightarrow{v}+\overrightarrow{v}.\overrightarrow{w}+\overrightarrow{w}.\overrightarrow{u}$

asked Apr 3, 2013 by poojasapani_1

1 answer

If$|\overrightarrow{a}+\overrightarrow{b}|=60, |\overrightarrow{a}-\overrightarrow{b}|=40$ and $|\overrightarrow{b}|=46$ find$|\overrightarrow{a}|.$

asked Apr 3, 2013 by poojasapani_1

1 answer

If $\overrightarrow{a}\;,\overrightarrow{b}\;,\overrightarrow{c}$ are three mutually perpendicular unit vectors,than prove that $|\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}|=\sqrt{3}$

asked Apr 3, 2013 by poojasapani_1

1 answer

If the sum of two unit vectors is a unit vector prove that the magnitude of their difference is $\sqrt{3}$.

asked Apr 3, 2013 by poojasapani_1

1 answer

Find the value of $m$ for which the vectors $\overrightarrow{a}=\overrightarrow{3i}+\overrightarrow{2j}+\overrightarrow{9k}$ and $\overrightarrow{b}=\overrightarrow{i}+\overrightarrow{mj}+\overrightarrow{3k}$ are perpendicular

asked Apr 3, 2013 by poojasapani_1

1 answer

If $\hat{a}$ and $\hat{b}$ are unit vectors inclined at an angle $\theta$ ,than prove that cos$\Large\frac{\theta}{2}$=$\Large\frac{1}{2}$$|\hat{a}$+$\hat{b}| $

asked Apr 3, 2013 by poojasapani_1

1 answer

If $\hat{a}$ and $\hat{b}$ are unit vectors inclined at an angle $\theta$ ,than prove that tan$\large\frac{\theta}{2}$=$\large\frac{|\hat{a}-\hat{b}|}{|\hat{a}+\hat{b}|}$

asked Apr 2, 2013 by poojasapani_1

1 answer

Show that the vector $\overrightarrow{i}+\overrightarrow{j}+\overrightarrow{k}$ is equally inclined with the coordinate axes.

asked Apr 2, 2013 by poojasapani_1

1 answer

Find the angles which the vector $\overrightarrow{i}-\overrightarrow{j}+\sqrt{2}\overrightarrow{k}$ makes with the coordinate axes.

asked Apr 2, 2013 by poojasapani_1

1 answer

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