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Questions >> CBSE XII >> Math >> Vector Algebra

If $\overrightarrow a . \overrightarrow a = 0$ and $\overrightarrow a . \overrightarrow b = 0$ , then what can be concluded about the vector $\overrightarrow b$ ?

asked Dec 6, 2012 by thanvigandhi_1

1 answer

If $\overrightarrow a, \overrightarrow b, \overrightarrow c,$ are unit vectors such that $\overrightarrow a + \overrightarrow b + \overrightarrow c = \overrightarrow 0$ , find the value of $\overrightarrow a ⋅ \overrightarrow b + \overrightarrow b ⋅ \overrightarrow c + \overrightarrow c ⋅ \overrightarrow a$ .

asked Dec 6, 2012 by thanvigandhi_1

1 answer

If either vector $\overrightarrow a = \overrightarrow 0$ or $\overrightarrow b = \overrightarrow 0$ , then $\overrightarrow a . \overrightarrow b = 0$ . But the converse need not be true. Justify your answer with an example.

asked Dec 6, 2012 by thanvigandhi_1

1 answer

If the vertices $A, B, C$ of a triangle $ABC$ are $(1, 2, 3), (-1, 0, 0), (0, 1, 2)$ respectively, then find $\angle ABC. \angle ABC$ is the angle between the vectors $\overline {BA}$ and $\overline {BC}$ .

asked Dec 6, 2012 by thanvigandhi_1

1 answer

Show that the points A(1, 2, 7), B(2, 6, 3) and C(3, 10, –1) are collinear.

asked Dec 6, 2012 by thanvigandhi_1

1 answer

Show that the vectors $2\hat i - \hat j + \hat k, \: \hat i - 3\hat j - 5\hat k$ and $3\hat i - 4\hat j - 4\hat k$ form the vertices of a right angled triangle.

asked Dec 6, 2012 by thanvigandhi_1

1 answer

If $\overrightarrow a$ is a nonzero vector of magnitude $" a "$ and $λ$ a nonzero scalar, then $λ \overrightarrow a$ is unit vector if

asked Dec 6, 2012 by thanvigandhi_1

1 answer

Find $| a × b |$ , if $a = \hat i - 7\hat j + 7\hat k$ and $b = 3\hat i - 2\hat j + 2\hat k$

asked Dec 5, 2012 by thanvigandhi_1

1 answer

Find a unit vector perpendicular to each of the vector $\overrightarrow a + \overrightarrow b$ and $\overrightarrow a - \overrightarrow b$ , where $\overrightarrow a = 3\hat i + 2\hat j + 2\hat k$ and $\overrightarrow b = \hat i + 2\hat j - 2\hat k$

asked Dec 5, 2012 by thanvigandhi_1

1 answer

If a unit vector $\overrightarrow a$ makes angles $\frac{\large \pi}{\large 3}$ with $\hat i, \frac{\large \pi}{\large 3}$ with $\hat j$ and an acute angle $θ$ with $\hat k$ , then find $θ$ and hence, the components of $\overrightarrow a$ .

asked Dec 5, 2012 by thanvigandhi_1

1 answer

Show that $( \overrightarrow a - \overrightarrow b) \times ( \overrightarrow a + \overrightarrow b) = 2 ( \overrightarrow a \times \overrightarrow b)$

asked Dec 5, 2012 by thanvigandhi_1

1 answer

Find $λ$ and $μ$ if $( 2\hat i + 6\hat j + 27\hat k)\times (\hat i + λ\hat j + μ\hat k) = \overrightarrow 0$

asked Dec 5, 2012 by thanvigandhi_1

1 answer

Given that $\overrightarrow a . \overrightarrow b = 0$ and $\overrightarrow a$ x $\overrightarrow b = \overrightarrow 0$ What can you conclude about the vectors $\overrightarrow a$ and $\overrightarrow b$ ?

asked Dec 5, 2012 by thanvigandhi_1

1 answer

Let the vectors $\overrightarrow a, \overrightarrow b, \overrightarrow c$ be given as $a_1\hat i + a_2\hat j + a_3\hat k, \: b_1\hat i + b_2\hat j + b_3\hat k, \: c_1\hat i + c_2\hat j + c_3\hat k$ . Then show that $\overrightarrow a × (\overrightarrow b + \overrightarrow c) = \overrightarrow a × \overrightarrow b + \overrightarrow a × \overrightarrow c$

asked Dec 5, 2012 by thanvigandhi_1

1 answer

If either $\overrightarrow a = \overrightarrow 0$ or $\overrightarrow b = \overrightarrow 0$ then $\overrightarrow a × \overrightarrow b = \overrightarrow 0$ . Is the converse true? Justify your answer with an example.

asked Dec 5, 2012 by thanvigandhi_1

1 answer

Find the area of the triangle with vertices $A(1, 1, 2), B(2, 3, 5)$ and $C(1, 5, 5)$ .

asked Dec 5, 2012 by thanvigandhi_1

1 answer

Find the area of the parallelogram whose adjacent sides are determined by the vectors $\overrightarrow a = \hat i - \hat j + 3\hat k$ and $\overrightarrow b = 2\hat i - 7 \hat j + \hat k$

asked Dec 5, 2012 by thanvigandhi_1

1 answer

Let the vectors $\overrightarrow a$ and $\overrightarrow b$ be such that $| \overrightarrow a | = 3$ and $|\overrightarrow b | = \frac{\large \sqrt 2}{\large 3}$ , then $\overrightarrow a \times \overrightarrow b$ is a unit vector, if the angle between $\overrightarrow a$ and $\overrightarrow b$ is

asked Dec 5, 2012 by thanvigandhi_1

1 answer

Area of a rectangle having vertices $A, B, C$ and $D$ with position vectors $-\hat i + \large\frac{1}{2}$ $\hat j + 4\hat k, \: \hat i +\large \frac{1}{2}$ $\hat j + 4\hat k, \: \hat i -\large \frac{1}{2}$ $\hat j + 4\hat k$ and $-\hat i -\large \frac{1}{2}$ $\hat j + 4\hat k$ respectively is

asked Dec 5, 2012 by thanvigandhi_1

1 answer

Write down a unit vector in XY plane, making an angle of 30 degress with the positive direction of $x$ axis.

asked Dec 5, 2012 by thanvigandhi_1

1 answer

Find the scalar components and magnitude of the vector joining the points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$

asked Dec 5, 2012 by thanvigandhi_1

1 answer

A girl walks 4 km towards west, then she walks 3 km in a direction 30° east of north and stops. Determine the girl’s displacement from her initial point of departure.

asked Dec 5, 2012 by thanvigandhi_1

1 answer

If $\overrightarrow a = \overrightarrow b + \overrightarrow c$ , then is it true that $| \overrightarrow a | = | \overrightarrow b | + | \overrightarrow c |$ ? Justify your answer.

asked Dec 5, 2012 by thanvigandhi_1

1 answer

Find the value of $x$ for which $x(\hat i + \hat j +\hat k)$ is a unit vector.

asked Dec 5, 2012 by thanvigandhi_1

1 answer

Find a vector of magnitude $5$ units, and parallel to the resultant of the vectors $\overrightarrow a = 2\hat i + 3 \hat j - \hat k$ and $\overrightarrow b = \hat i - 2\hat j + \hat k$

asked Dec 5, 2012 by thanvigandhi_1

1 answer

If $\overrightarrow a = \hat i + \hat j + \hat k, \overrightarrow b = 2\hat i - \hat j + 3\hat k$ and $\overrightarrow c = \hat i - 2\hat j + \hat k$ , find a unit vector parallel to the vector $2\overrightarrow a - \overrightarrow b + 3\overrightarrow c$

asked Dec 5, 2012 by thanvigandhi_1

1 answer

Show that the points A(1, – 2, – 8), B (5, 0, – 2) and C (11, 3, 7) are collinear, and find the ratio in which B divides AC.

asked Dec 5, 2012 by thanvigandhi_1

1 answer

Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are $(2\overrightarrow a + \overrightarrow b)$ and $(\overrightarrow a - 3\overrightarrow b)$ externally in the ratio 1 : 2. Also, show that P is the mid point of the line segment RQ

asked Dec 5, 2012 by thanvigandhi_1

1 answer

The two adjacent sides of a parallelogram are $2\hat i - 4\hat j + 5\hat k$ and $\hat i - 2\hat j - 3\hat k$ . Find the unit vector parallel to its diagonal. Also, find its area.

asked Dec 5, 2012 by thanvigandhi_1

1 answer

Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ are $\frac{\large 1}{\large \sqrt 3}, \: \frac{\large 1}{\large \sqrt 3}, \: \frac{\large 1}{\large \sqrt 3}$

asked Dec 5, 2012 by thanvigandhi_1

1 answer

Let $\overrightarrow a = \hat i + 4 \hat j + 2\hat k, \overrightarrow b = 3\hat i − 2 \hat j + 7\hat k \: and \: \overrightarrow c = 2\hat i − \hat j + 4\hat k$ . Find a vector $\overrightarrow d$ which is perpendicular to both $\overrightarrow a$ and $\overrightarrow b$ , and $\overrightarrow c ⋅ \overrightarrow d =15$

asked Dec 5, 2012 by thanvigandhi_1

1 answer

The scalar product of the vector $\hat i + \hat j + \hat k$ with a unit vector along the sum of vectors $2\hat i + 4 \hat j − 5\hat k$ and $λ\hat i + 2 \hat j + 3\hat k$ is equal to one. Find the value of $λ.$

asked Dec 5, 2012 by thanvigandhi_1

1 answer

If $\overrightarrow a, \overrightarrow b, \overrightarrow c,$ are mutually perpendicular vectors of equal magnitudes, show that the vector $\overrightarrow a + \overrightarrow b + \overrightarrow c$ is equally inclined to $\overrightarrow a, \overrightarrow b and \overrightarrow c$ .

asked Dec 5, 2012 by thanvigandhi_1

1 answer

Prove that $(\overrightarrow a + \overrightarrow b) ⋅ (\overrightarrow a + \overrightarrow b) =| \overrightarrow a |^2 + |\overrightarrow b |^2$ , if and only if $\overrightarrow a,\: \overrightarrow b$ are perpendicular, given $\overrightarrow a ≠ 0, \overrightarrow b ≠ 0$ .

asked Dec 5, 2012 by thanvigandhi_1

1 answer

Choose the correct answer if θ is the angle between two vectors $\overrightarrow a$ and $\overrightarrow b$ , then $\overrightarrow a \: . \: \overrightarrow b\ \: \geq 0$ only when

asked Dec 5, 2012 by thanvigandhi_1

1 answer

Choose the correct answer in the following : let $\overrightarrow a$ and $\overrightarrow b$ be two unit vectors and θ is the angle between them. Then $\overrightarrow a\ + \overrightarrow b$ is a unit vector if

asked Dec 5, 2012 by thanvigandhi_1

1 answer

Choose the correct answer if the value of $\hat i.( \hat j\times \hat k)+ \hat j . (\hat i \times \hat k) +\hat k. (\hat i \times \hat j)$ is

asked Dec 5, 2012 by thanvigandhi_1

1 answer

Choose the correct answer if θ is the angle between any two vectors $\overrightarrow a$ and $\overrightarrow b$ , then $| \overrightarrow a ⋅\overrightarrow b | \: = \: | \overrightarrow a \times\overrightarrow b |$ when θ is equal to

asked Dec 5, 2012 by thanvigandhi_1

1 answer

What is the value of $\hat{i} (\hat{j}. \hat{k}) \hat{j} (\hat{i}. \hat{k}) \hat{k} (\hat{j}. \hat{i})?$

asked Nov 2, 2012 by balaji.thirumalai

1 answer

If $\theta$ is the angle between any two vectors $\overrightarrow{a} and \overrightarrow{b} ,$ then $|\overrightarrow{a}.\overrightarrow{b} | = | \overrightarrow{a} \times \overrightarrow{b} |$ when $\theta$ is equal to

asked Nov 2, 2012 by balaji.thirumalai

1 answer

If θ is the angle between two vectors $\overrightarrow{a} and \overrightarrow{b}$ when is $\overrightarrow{a} ⋅ \overrightarrow{b} ≥ 0$

asked Nov 2, 2012 by balaji.thirumalai

1 answer

Prove that $(\overrightarrow{a} + \overrightarrow{b} ) ⋅ (\overrightarrow{a} + \overrightarrow{b} ) =| \overrightarrow{a} |2 + | \overrightarrow{b} |2$ if and only if $\overrightarrow{a} , \overrightarrow{b}$ are perpendicular, given $\overrightarrow{a} ≠ 0, \overrightarrow{b} ≠ 0$

asked Nov 2, 2012 by balaji.thirumalai

0 answers

If $\overrightarrow{a} ,\overrightarrow{b} , \overrightarrow{c}$ are mutually perpendicular vectors of equal magnitudes, show that the vector $\overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c}$ is equally inclined to $\overrightarrow{a}, \overrightarrow{b} and \overrightarrow{c}$ .

asked Nov 2, 2012 by balaji.thirumalai

0 answers

Given $\overrightarrow{a}, \overrightarrow{b} and \overrightarrow{c}$ Find a vector $\overrightarrow{d}$ which is perpendicular to both $\overrightarrow{a} and \overrightarrow{b}, and \overrightarrow{c} ⋅\overrightarrow{d} = 15$

asked Nov 2, 2012 by anonymous

0 answers

Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ are $1 / \sqrt{3}, 1 / \sqrt{3}$ and $1 / \sqrt{3}$

asked Nov 2, 2012 by balaji.thirumalai

0 answers

The two adjacent sides of a parallelogram are given. Find the unit vector parallel to its diagonal. Also, find its area.

asked Nov 2, 2012 by balaji.thirumalai

0 answers

Find the position vector of a point tand show that P is the mod point of the line segment

asked Nov 2, 2012 by balaji.thirumalai

0 answers

Show that the points A (1, – 2, – 8), B (5, 0, – 2) and C (11, 3, 7) are collinear, and find the ratio in which B divides AC.

asked Nov 2, 2012 by balaji.thirumalai

0 answers

Find a unit vector parallel to the vector....

asked Nov 2, 2012 by balaji.thirumalai

0 answers

Find a vector of magnitude 5 units, and parallel to the resultant of the vectors

asked Nov 2, 2012 by balaji.thirumalai

0 answers

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