Represent the following graphically:

i. a displacement of 40 km, 30° east of north

ii. a displacement of 50 km south - east

iii. a displacement of 70 km, 40° north of west.

Classify the following measures as scalars and vectors :

i. 15 kg

ii. 20 kg weight

iii. 45°

iv. 10 meters south - east

v. 50 m/sec²

Classify the following as scalars and vector quantities:

i. Time period

ii. Distance

iii. Displacement

iv. Force

v. Work

vi. Velocity

vii. Acceleration

In fig 23.5 ABCD is a regular hexagon, which vectors are:

i. Collinear

ii. Equal

iii. Cointitial

iv. Collinear but not equal.

Answer the following as true or false:

If P, Q and R are three collinear points such that and Find the vector .

Give a condition that three vectors and form the three sides of a triangle. What are the other possibilities?

If and are two non-collinear vectors having the same initial point. What are the vectors represented by and

If is a vector and m is a scalar such that m then what are the alternatives for m and ?

If are two vectors, then write the truth value of the following statements :

If are two vectors, then write the truth value of the following statements :

If are two vectors, then write the truth value of the following statements :

ABCD is a quadrilateral. Find the sum of the vectors and

ABCDE is a pentagon, prove that

ABCDE is a pentagon, prove that

Prove that the sum of all vectors drawn from the centre of a regular octagon to its vertices is the zero vector.

If P is a point and ABCD is a quadrilateral and show that ABCD is a parallelogram.

Five forces and and act at the vertex of a regular hexagon ABCDEF. Prove that the resultant is 6 where O is the centre of hexagon.

Let be the position vectors of the four distinct points A, B, C, D. If then show that ABCD is a parallelogram.

If are the position vectors of A, B respectively, find the position vector of a point C in AB produced such that AC = 3 AB and that a point D in BA produced such that BD = 2 BA.

Show that the four points A, B, C, D with position vectors respectively such that are coplanar. Also, find the position vector of the point of intersection of the line segments AC and BD.

Show that the four points P, Q, R, S with position vectors respectively such that are coplanar. Also, find the position vector of the point of intersection of the line segments PR and QS.

The vertices A, B, C of triangle ABC have respectively position vectors with respect to a given origin O. Show that the point D where the bisector of ∠A meets BC has position vector where and

If O is a point in space, ABC is a triangle and D, E, F are the mid-points of the sides BC, CA and AB respectively of the triangle, prove that

Show that the sum of the three vectors determined by the medians of a triangle directed from the vertices is zero.

ABCD is a parallelogram and P is the point of intersection of its diagonals. If O is the origin of reference, show that

Show that the line segments joining the midpoints of opposite sides of a quadrilateral bisect each other.

ABCD are four points in a plane and Q is the point of intersection of the lines joining the mid-points of AB and CD ; BC and AD. Show that where P is any point.

Prove by vector method that the internal bisectors of the angles of a triangle are concurrent.

If the position vector of a point (–4, –3) be find

If the position vector of a point (12, n) is such that find the value(s) of n.

Find a vector of magnitude 4 units which is parallel to the vector

Express in terms of unit vectors and when the points are :

(i) A (4, -1), B(1, 3)

(ii) A(-6, 3), B(-2, -5)

Find in each case.

Find the coordinates of the tip of the position vector which is equivalent to where the coordinates of A and B are (-1, 3) and (-2, 1) respectively.

ABCD is a parallelogram. If the coordinates of A, B and C are (–2, 1), (3, 0) and (1, –2), find the coordinates of D.

If the position vectors of the points A (3, 4), B (5, -6) and C (4, -1) are respectively, compute

If be the position vector whose tip is (5, -3), find the coordinates of a point B such that the coordinates of A being (4, -1).

Show that the points and form an isosceles triangle.

Find a unit vector parallel to the vector

The position vectors of points A, B and C are and respectively. If C divides the lien segment joining A and B in the ratio 3 : 1, find the values of λ and μ.

Find the components along the coordinate axes of the position vector of each of the following points –

i. P (3, 2)

ii. Q (5, 1)

iii. R (–11, –9)

iv. S (4, –3)

Find the unit vector in the direction of

Find a unit vector in the direction of the resultant of the vectors and

The adjacent sides of a parallelogram are represented by the vectors and Find unit vectors parallel to the diagonals of the parallelogram.

If and find

If and the coordinates of P are (1, -1, 2), find the coordinates of Q.

Prove that the points and are the vertices of a right-angled triangle.

If the vertices A, B, C of a triangle ABC are the points with position vectors respectively, what are the vectors determined by its sides? Find the length of these vectors.

Find the vector from the origin O to the centroid of the triangle whose vertices are (1, -1, 2), (2, 1, 3) and (-1, 2, -1).

Find the position vector of a point R which divides the line segment joining points and in the ratio 2 : 1.

(i) Internally

(ii) Externally

Find the position vector of the mid-point of the vector joining the points and

Find the unit vector in the direction of vector where P and Q are the points (1, 2, 3) and (4, 5, 6).

Show that the points are the vertices of a right angled triangle.

Find the position vector of the mid-point of the vector joining the points P (2, 3, 4) and Q (4, 1, -2).

Find the value of x for which is a unit vector.

If and find a unit vector parallel to

If and find a vector of magnitude 6 units which is parallel to the vector

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

The two vectors and represent the sides and respectively of triangle ABC. Find the length of the median through A.

If are non-coplanar vectors, prove that the points having the following position vectors are collinear:

If are non-coplanar vectors, prove that the points having the following position vectors are collinear:

Prove that the points having position vectors are collinear.

If the points with position vectors and are collinear, find the value of a.

If are two non-collinear vectors, prove that the points with position vectors and are collinear for all real values of

If prove that A, B, C are collinear points.

Show that the vectors and are collinear.

If the points A (m, –1), B (2, 1) and C(4, 5) are collinear, find the value of m.

Show that the points (3, 4), (–5, 16), (5, 1) are collinear.

If the vectors and are collinear, find the value of m.

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.

Using vectors show that the points
A(–2, 3, 5), B(7, 0, 1) C(–3, –2, –5) and D(3, 4, 7) are such that AB and CD intersect at the point P(1, 2, 3).

Using vectors, find the value of such that the points (1, –1, 3) and (3, 5, 3) are collinear.

Show that the points whose position vectors are as given below are collinear :

and

Show that the points whose position vectors are as given below are collinear :

and

Using vector method, prove that the following points are collinear.

A(6, –7, –1), B(2 –3, 1) and C(4, –5, 0)

Using vector method, prove that the following points are collinear.

A(2, –1, 3), B(4, 3, 1) and C(3, 1, 2)

Using vector method, prove that the following points are collinear.

A(1, 2, 7), B(2, 6, 3) and C(3, 10 –1)

Using vector method, prove that the following points are collinear.

A(–3, –2, –5), B(1, 2, 3) and C(3, 4, 7)

Using vector method, prove that the following points are collinear.

A (2, –1, 3), B (3, –5, 1) and C (–1, 11, 9).

If are non–zero, non-coplanar vectors, prove that the following vectors are coplanar :

and

If are non–zero, non-coplanar vectors, prove that the following vectors are coplanar :

and

Show that the four points having position vectors are coplanar.

Prove that the following vectors are coplanar :

and

Prove that the following vectors are coplanar :

and

Prove that the following vectors are non-coplanar :

and

Prove that the following vectors are non–coplanar :

and

If are non–coplanar vectors, prove that the following vectors are non–coplanar :

and

If are non–coplanar vectors, prove that the following vectors are non-coplanar :

and

Show that the vectors given
by and are non-coplanar. Express vector as a linear combination of the vectors and

Prove that a necessary and sufficient condition for three vectors and to be coplanar is that there exist scalars not all zero simultaneously such that

Show that the four points A, B, C and D with position vectors and respectively are coplanar if and only if

Can a vector have direction angles 45^o, 60^o, 120^o.

Prove that 1, 1, and 1 cannot be direction cosines of a straight line.

A vector makes an angle of π/4 with each of x - axis and y - axis. Find the angle made by it with the z - axis.

A vector is inclined to the x - axis at 45^o and y - axis at 60^o. If units, find

Find the direction cosines of the following vectors :

i.

ii.

iii.

Find the angles at which the following vectors are inclined to each of the coordinate axes :

i.

ii.

iii.

Show that the vector is equally inclined with the axes OX, OY and OZ.

Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ are

If a unit vector makes an angle with with and an acute angle θ with then find θ and hence, the components of

Find a vector of magnitude units which makes an angle of and with y and z - axes respectively.

A vector is inclined at equal angles to the three axes. If the magnitude of is find

Define “zero vector”.

Define unit vector.

Define position vector of a point.

If are two non-collinear vectors such that , then write the values of x and y.

If represent two adjacent sides of a parallelogram, then write vectors representing its diagonals.

If represent the sides of a triangle taken in order, then write the value of

If are position vectors of the vertices A, B and C respectively, of a triangle ABC, write the value of .

If are position vectors of the points A, B and C respectively, write the value of .

If are the position vectors of the vertices of a triangle, then write the position vector of its centroid.

If G denotes the centroid of Δ ABC, then write the value of

If denote the position vectors of points A and B respectively and C is a point on AB such that 3AC = 2AB, then write the position vector of C.

If D is the mid-point of side BC of a triangle ABC such that write the value of λ.

If D, E, F are the mid-points of the sides BC, CA and AB respectively of a triangle ABC, write the value of

If is non-zero vector of modulus a and m is a non-zero scalar such that m is a unit vector, write the value of m.

If are the position vectors of the vertices of an equilateral triangle whose orthocentre is at the origin, then write the value of .

Write a unit vector making equal acute angles with a coordinates axes.

If a vector makes angles α, β, γ with OX, OY and OZ respectively, then write the value of sin²α + sin²β + sin²γ.

Write a vector of magnitude 12 units which makes 45° angle with X-axis, 60° angle with y-axis and an obtuse angle with Z-axis.

Write the length (magnitude) of a vector whose projections on the coordinate axes are 12, 3 and 4 units.

Write the position vector of a point dividing the line segment joining points A and B with position vectors externally in the ratio 1 : 4, where and

Write the direction cosines of the vector

If write unit vectors parallel to

If write a unit vector along the vector

Write the position vector of a point dividing the line segment joining points having position vectors externally in the ratio 2 : 3.

If fine the unit vector in the direction of

If and find

A unit vector makes angles respectively and a acute angle θ with Find θ.

Write a unit vector in the direction of

If and find a unit vector parallel to

Write a unit vector in the direction of

Find the position vector of the mid-point of the line segment AB, where A is the point (3, 4, –2) and B is the point (1, 2, 4).

Find a vector in the direction of which has magnitude of 6 units.

What is the cosine of the angle which the vector makes with y-axis?

Write two different vectors having same magnitude.

Write two different vectors having same direction.

Write a vector in the direction of vector which has magnitude of 8 unit.

Write the direction cosines of the vector

Find a unit vector in the direction of

For what value of ‘a’ the vectors and are collinear?

Write the direction cosines of the vectors

Find the sum of the following vectors:

Find a unit vector in the direction of the vector

If and are two equal vectors, then write the value of x + y + z.

Write a unit vector in the direction of the sum of the vectors and

Find the value of ‘p’ for which the vectors and are parallel.

Find a vector of magnitude making an angle of π/4 with x-axis π/2 with y-axis and an acute angle θ with z-axis.

Write a unit vector in the direction of , where P and Q are the points (1, 3, 0) and (4, 5, 6) respectively.

Find a vector in the direction of vector which has magnitude 21 units.

If and –3 ≤ λ ≤ 2, then write the range of

In a triangle OAC, if B is the mid-point of side AC and then what is ?

Write the position vector of the point which divides the join of points with position vectors and in the ratio 2:1.

Mark the correct alternative in each of the following:

If in a Δ ABC, A ≡ (0, 0), B ≡ (3, 3, √3), C ≡ (–3, √3, 3), then the vector of magnitude units directed along AO, where O is the circumcentre of Δ ABC is

Mark the correct alternative in each of the following:

If are the vectors forming consecutive sides of a regular hexagon ABCDEF, then the vector representing side CD is

Mark the correct alternative in each of the following:

Forces act along OA and OB. If their resultant passes through C on AB, then

Mark the correct alternative in each of the following:

If are three non-zero vectors, no two of which are collinear and the vector is collinear with is collinear with , then

Mark the correct alternative in each of the following:

If and points are collinear, then a is equal to

Mark the correct alternative in each of the following:

If G is the intersection of diagonals of a parallelogram ABCD and O is any point, then

Mark the correct alternative in each of the following:

The vector is a

Mark the correct alternative in each of the following:

In a regular hexagon ABCDEF, Then,

Mark the correct alternative in each of the following:

The vector equation of the plane passing through is provided that

Mark the correct alternative in each of the following:

If O and O’ are circumcentre and orthocentre of Δ ABC, then equals

Mark the correct alternative in each of the following:

If are the position vectors of points A, B, C, D such that no three of them are collinear and then ABCD is a

Mark the correct alternative in each of the following:

Let G be the centroid of Δ ABC. If then the bisector in terms of is

Mark the correct alternative in each of the following:

If ABCDEF is a regular hexagon, then equals.

Mark the correct alternative in each of the following:

The position vectors of the points A, B, C are and respectively. These points

Mark the correct alternative in each of the following:

If three points A, B and C have position vectors and respectively are collinear, then (x, y) =

Mark the correct alternative in each of the following:

ABCD is a parallelogram with AC and BD as diagonals. Then,

Mark the correct alternative in each of the following:

If OACB is a parallelogram with and , then

Mark the correct alternative in each of the following:

If are two collinear vectors, then which of the following are incorrect?

Mark the correct alternative in each of the following:

If figure which of the following is not true?