Chapter 10
Write down a unit vector in XY-plane, making an angle of 30° with the positive direction of x-axis.
If a parabolic reflector is 20 cm in diameter and 5 cm deep, find the focus.
The two adjacent sides of a parallelogram are ˆ ˆˆ ˆ ˆ ˆ2 4 5 and 2 3i j k i j k− + − − . Find the unit vector parallel to its diagonal. Also, find its area.
Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ are ± 1/3 1/3 1/3 , , .
Let . Find a vector which is perpendicular to both and , and .
The scalar product of the vector ˆˆ ˆi j k+ + with a unit vector along the sum of vectors ˆˆ ˆ2 4 5i j k+ − and ˆˆ ˆ 2 3i j kλ + + is equal to one. Find the value of λ .
If are mutually perpendicular vectors of equal magnitudes, show that the vector is equally inclined to .
Prove that , if and only if are perpendicular, given . Choose the correct answer in Exercises 16 to 19.
If θ is the angle between two vectors , then only when (A) 0 2 π< θ < (B) 0 2 π≤ θ ≤ (C) 0 < θ < π (D) 0 ≤ θ ≤ π
Let be two unit vectors and θ is the angle between them. Then is a unit vector if (A) / πθ = (B) / πθ = (C) πθ = (D) 2 πθ =
The value of ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ.( ) ( ) ( )i j k j i k k i j× + ⋅ × + ⋅ × is (A) 0 (B) -1 (C) 1 (D) 3
If θ is the angle between any two vectors , then when θ is equal to (A) 0 (B) 4 / π (C) 2 / π (D) π
An arch is in the form of a parabola with its axis vertical. The arch is 10 m high and 5 m wide at the base. How wide is it 2 m from the vertex of the parabola?
Find the scalar components and magnitude of the vector joining the points P(x1, y1, z1) and Q (x2, y2, z2).
The cable of a uniformly loaded suspension bridge hangs in the form of a parabola. The roadway which is horizontal and 100 m long is supported by vertical wires attached to the cable, the longest wire being 30 m and the shortest being 6 m. Find the le ngth of a supporting wire attached to the roadway 18 m from the middle.
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.
An arch is in the form of a semi-ellipse. It is 8 m wide and 2 m high at the centre. Find the height of the arch at a point 1.5 m from one end.
If , then is it true that ? Justify your answer.
A rod of length 12 cm moves with its ends always touching the coordinate axes. Determine the equation of the locus of a point P on the rod, which is 3 cm from the end in contact with the x-axis.
Find the value of x for which ˆˆ ˆ( )x i j k+ + is a unit vector.
Find a vector of magnitude 5 units, and parallel to the resultant of the vectors .
Find the area of the triangle formed by the lines joining the vertex of the parabola x2 = 12y to the ends of its latus rectum.
If , find a unit vector parallel to the vector .
A man running a racecourse notes that the sum of the distances from the two flag posts from him is always 10 m and the distance between the flag posts is 8 m. Find the equation of the posts traced by the man.
An equilateral triangle is inscribed in the parabola y2 = 4 ax, where one vertex is at the vertex of the parabola. Find the length of the side of the triangle.
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.
Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are externally in the ratio 1 : 2. Also, show that P is the mid point of the line segment RQ.