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modern engineering mathematics
Questions and Answers of
Modern Engineering Mathematics
Find a unit vector perpendicular to the plane of the two vectors (2, –1, 1) and (3, 4, –1). What is the sine of the angle between these two vectors?
Prove that the shortest distance of a point P from the line through the points A and B isA satellite is stationary at P(2, 5, 4) and a warning signal is activated if any object comes within a
The position vector r, with respect to a given origin O, of a charged particle of mass m and charge e at time t is given bywhere E, B, a and ω are constants. The corresponding velocity and
Find the volume of the parallelepiped whose edges are represented by the vectors (2, –3, 4), (1, 3, –1), (3, –1, 2).
Prove that the vectors (3, 2, –1), (5, –7, 3) and (11, –3, 1) are coplanar.
Find the constant λ such that the three vectors (3, 2, –1), (1, –1, 3) and (2, –3, λ) are coplanar.
Prove that the four points having position vectors (2, 1, 0), (2, –2, –2), (7, –3, –1) and (13, 3, 5) are coplanar.
Given p = (1, 4, 1), q = (2, 1, –1) and r = (1, –3, 2), find(a) A unit vector perpendicular to the plane containing p and q;(b) A unit vector in the plane containing p × q and p × r that has
Show that if a is any vector and û any unit vector thenand draw a diagram to illustrate this relation geometrically. The vector (3, –2, 6) is resolved into two vectors along and perpendicular to
Three vectors u, v, w are expressed in terms of the three vectors l, m, n in the formShow thatand evaluate λ.
Forces F1, F2, . . . . , Fn act at the points r1, r2, . . . , rn respectively. The total force and the total moment about the origin O areShow that for any other origin O´ the moment is given byfind
Extended exercise on products of four vectors.(a) Use (4.11) to showand use (4.13) to simplify the expression on the right-hand side.(b) Use (4.13) to show thatand show that the right-hand side can
If A and B have position vectors (1, 2, 3) and (4, 5, 6) respectively, find(a) The direction vector of the line through A and B;(b) The vector equation of the line through A and B;(c) The cartesian
Find the vector equation of the line through the point A with position vector OA(vector) = (2, 1, 1) in the direction d = (1, 0, 1). Does this line pass through any of the points (1, 1, 0), (1, 1,
Show that the line joining (2, 3, 4) to (1, 2, 3) is perpendicular to the line joining (1, 0, 2) to (2, 3, –2).
Prove that the lines r = (1, 2, –1) + t(2, 2, 1) and r = (–1, –2, 3) + s(4, 6, –3) intersect, and find the coordinates of their point of intersection. Also find the acute angle between the
P is a point on a straight line with position vector r = a + tb. Show thatBy completing the square, show that r2 is a minimum for the point P for which t = –a · b/b2. Show that at this point
Find the vector equation of the line through the points with position vectors a = (2, 0, –1) and b = (1, 2, 3). Write down the equivalent cartesian coordinate form. Does this line intersect the
Find the shortest distance between the two lines
Find the vector equation of the plane that passes through the points (1, 2, 3), (2, 4, 5) and (4, 5, 6). What is its cartesian equation?
Find the equation of the plane with perpendicular n = (1, –1, 1) that passes through the point with position vector (2, 3, 3). Show that the line with equation r = (–1, –1, 2) + t(2, 0, –2)
Find the vector equation of the plane that contains the line r = a + λb and passes through the point with position vector c.
The line of intersection of two planes r · n1 = p1 and r · n2 = p2 lies in both planes. It is therefore perpendicular to both n1 and n2. Give an expression for this direction, and so show that the
Find the equation of the line through the point (1, 2, 4) and in the direction of the vector (1, 1, 2). Find where this line meets the plane x + 3y – 4z = 5.
Find the acute angle between the planes 2x + y – 2z = 5 and 3x – 6y – 2z = 7.
Given that a = (3, 1, 2) and b = (1, –2, –4) are the position vectors of the points P and Q respectively, find(a) The equation of the plane passing through Q and perpendicular to PQ;(b) The
Find the equation of the line joining (1, –1, 3) to (3, 3, –1). Show that it is perpendicular to the plane 2x + 4y – 4z = 5, and find the angle that the line makes with the plane 12x – 15y +
Find the equation of the plane through the lineand parallel to the line
Find the equation of the line through P(–1, 0, 1) that cuts the line r = (3, 2, 1) + t(1, 2, 2) at right angles at Q. Also find the length PQ and the equation of the plane containing the two lines
Show that the equation of the plane through the points P1, P2 and P3 with position vectors r1, r2 and r3 respectively takes the form
If P has coordinates (2, –1, 3), find the length OP and the direction cosines of OP.
A surveyor sets up her theodolite on horizontal ground, at a point O, and observes the top of a church spire, as illustrated in Figure 4.3. Relative to axes Oxyz, with Oz vertical, the surveyor
A cyclist travels at a steady 16 km h–1 on the four legs of his journey. From his origin, O, he travels for one hour in a NE direction to the point A; he then travels due E for half an hour to
From Figure 4.14, evaluate Figure 4.14
A quadrilateral OACB is defined in terms of the vectorsCalculate the vector representing the other two sides BC(vector) and CA(vector).
A force F has magnitude 2 N and a second force F´ has magnitude 1 N and is inclined at an angle of 60° to F, as illustrated in Figure 4.15. Find the magnitude of the resultant force R and the angle
An aircraft is flying at 400 knots in a strong NW wind of 50 knots. The pilot wishes to fly due west. In which direction should the pilot fly the aircraft to achieve this end, and what will be his
If ABCD is any quadrilateral, show thatwhere E and F are the midpoints of AB and DC respectively, and thatwhere X and Y are the midpoints of the diagonals AC and BD respectively.
Determine whether constants α and β can be found to satisfy the vector equationsand interpret the results.
Given the vectors a = (1, 1, 1), b = (–1, 2, 3) and c = (0, 3, 4), find(a) a + b(b) 2a – b(c) a + b – c(d) The unit vector in the direction of c.
Given a = (2, –3, 1) = 2i – 3j + k, b = (1, 5, –2) = i + 5j – 2k and c = (3, –4, 3) = 3i – 4j + 3k(a) Find the vector d = a – 2b + 3c;(b) Find the magnitude of d and write down a unit
A molecule XY3 has a tetrahedral form; the position vector of the X atom is (2√3 + √2, 0, –2 + √6) and those of the three Y atoms are
Three forces, with units of newtons, F1 = (1, 1, 1) F2 has magnitude 6 and acts in the direction (1, 2, –2) F3 has magnitude 10 and acts in the direction (3, –4, 0) act on a particle. Find the
Two geostationary satellites have known positions (0, 0,h) and (0, A, H) relative to a fixed set of axes on the Earth’s surface (which is assumed flat, with the x and y axes lying on the surface
A square is formed in the first and second quadrant with OP as one side of the square and OP(vector) = (1, 2). Find the coordinates of the other two vertices of the square.
M is the centre of a square with vertices A, B, C and D taken anticlockwise in that order. If, in the Argand diagram, M and A are represented by the complex numbers –2 + j and 1 + j5 respectively,
Given the vectors a = (1, –1, 2), b = (–2, 0, 2) and c = (3, 2, 1), evaluate
Find the angle between the vectors a = (1, 2, 3) and b = (2, 0, 4).
Given a = (1, 0, 1) and b = (0, 1, 0), show that a · b = 0, and interpret this result.
The three vectorsare given. Show that a · b = a · c and interpret the result.
In a triangle ABC show that the perpendiculars from the vertices to the opposite sides intersect in a point.
Find the work done by the force F = (3, –2, 5) in moving a particle from a point P to a point Q having position vectors (1, 4, –1) and (–2, 3, 1) respectively.
Find the component of the vector F = (2, –1, 3) in(a) The i direction(b) The direction(c) The direction (4, 2, –1)
Given the vectors a = (2, 1, 0), b = (2, –1, 1) and c = (0, 1, 1), evaluate
Find a unit vector perpendicular to the plane of the vectors a = (2, –3, 1) and b = (1, 2, –4).
Find the area of the triangle having vertices at P(1, 3, 2), Q(–2, 1, 3) and R(3, –2, –1).
Four vectors are constructed corresponding to the four faces of a tetrahedron. The magnitude of a vector is equal to the area of the corresponding face and its direction is the outward perpendicular
A force of 4 units acts through the point P(2, 3, –5) in the direction of the vector (4, 5, –2). Find its moment about the point A(1, 2, –3). See Figure 4.41. What are the moments of the force
A rigid body is rotating with an angular velocity of 5 rad s–1 about an axis in the direction of the vector (1, 3, –2) and passing through the point A(2, 3, –1). Find the linear velocity of the
A trapdoor is raised and lowered by a rope attached to one of its corners. The rope is pulled via a pulley fixed to a point A, 50 cm above the hinge, as shown in Figure 4.42.If the trapdoor is
Find λ so that a = (2, –1, 1), b = (1, 2, –3) and c = (3, λ, 5) are coplanar.
In a triangle OAB the sides OA (vector) = a and OB(vector) = b are given. Find the point P, with c = OP(vector), where the perpendicular bisectors of the two sides intersect. Hence prove that the
Three non-zero, non-parallel and non-coplanar vectors a, b and c are given. Three further vectors are written in terms of a, b and c as
If a = (3, –2, 1), b = (–1, 3, 4) and c = (2, 1, –3), confirm that
Verify that a × (b × c) ≠ (a × b) × c for the three vectors a = (1, 0, 0), b = (–1, 2, 0) and c = (1, 1, 1).
The vectors a, b and c and the scalar p satisfy the equations a · b = p and a × b × c and a is not parallel to b. Solve for a in terms of the other quantities and give a geometrical
Find the equation of the lines L1 through the points (0, 1, 0) and (1, 3, –1) and L2 through (1, 1, 1) and (–1, –1, 1). Do the two lines intersect and, if so, at what point?
The position vectors of the points A and B are (1, 4, 6) and (3, 5, 7) Find the vector equation of the line AB and find the points where the line intersects the coordinate planes.
The line L1 passes through the points with position vectors (5, 1, 7) and (6, 0, 8) and the line L2 passes through the points with position vectors (3, 1, 3) and (–1, 3, α) Find the value of α
A tracking station observes an aeroplane at two successive times to be (–500, 0, 1000) and (400, 400, 1050) relative to axes x in an easterly direction, y in a northerly direction and z vertically
It is necessary to drill to an underground pipeline in order to undertake repairs, so it is decided to aim for the nearest point from the measuring point. Relative to axes x, y in the horizontal
Find the shortest distance between the two skew linesAlso determine the equation of the common perpendicular. (Note that two lines are said to be skew if they do not intersect and are not parallel.)
A box with an open top and unit side length is observed from the direction (a, b, c), as in Figure 4.51. Determine the part of OC that is visible.
Find the equation of the plane through the three points a = (1, 1, 1), b = (0, 1, 2) and c = (–1, 1, –1)
A metal has a simple cubic lattice structure so that the atoms lie on the lattice points given by r = a(l, m, n) where a is the lattice spacing and l, m, n are integers. The metallurgist needs to
Find the point where the plane r · (1, 1, 2) = 3 meets the line r = (2, 1, 1) + λ(0, 1, 2)
Find the equation of the line of intersection of the two planes x + y + z = 5 and 4x + y + 2z = 15.
Triangles ABC and XYZ are shown in the following diagram, with BC = 32 cm, YZ = 8 cm and XY = 10 cm. Determine the length of AC.
(a) Convert the angle 104° to radians expressing the answer as a multiple of π and as a decimal.(b) Convert the angle 2π/3 from radians to degrees.
Find the value of the angle X° in each of the following diagrams:
A triangle has sides of lengths 13 cm, 13 cm and 10 cm. What is the area of the triangle?
1) Triangles ABC and XYZ are shown in the following diagram, with BC = 16 cm and YZ = 8 cm. If the area of triangle XYZ is 10 cm2, determine the area of triangle ABC.2) In the diagram below, AC and
Find the value of the angles X° and Y° in the following diagrams:
Given that a = 3i – j – 4k, b = –2i + 4j – 3k and c = i + 2j – k, find(a) The magnitude of the vector a + b + c(b) A unit vector parallel to 3a – 2b + 4c;(c) The angles between the
If the vertices X, Y and Z of a triangle have position vectors x = (2, 2, 6), y = (4, 6, 4) and z = (4, 1, 7) relative to the origin O, find(a) The midpoint of the side XY of the triangle;(b) The
The vertices of a tetrahedron are the pointsDetermine(a) The vectors WX(vector) and WY(vector);(b) The area of the face WXZ;(c) The volume of the tetrahedron WXZY;(d) The angles between the faces
Given a = (–1, –3, –1), b = (q, 1, 1) and c = (1, 1, q) determine the values of q for which(a) A is perpendicular to b(b) a × (b × c) = 0
Given the vectors a = (2, 1, 2) and b = (–3, 0, 4), evaluate the unit vectors â and b̂. Use these unit vectors to find a vector that bisects the angle between a and b.
A triangle, ABC, is inscribed in a circle, centre O, with AOC as a diameter of the circle. Take OA(vector) = a and OB = b. By evaluating AB(vector) ·CB (vector) show that angle ABC is a right angle.
According to the inverse square law, the force on a particle of mass m1 at the point P1 due to a particle of mass m2 at the point P2 is given byParticles of mass 3m, 3m, m are fixed at the points
Show that the vector a which satisfies the vector equationmust take the form a = (α, 2α –1, 1). If in addition the vector a makes an angle cos−1(1/3) with the vector (i – j + k) show that
The electric field at a point having position vector r, due to a charge e at R, is e(r – R)/|r – R|3. Find the electric field E at the point P(2, 1, 1) given that there is a charge e at each of
Given thatare the position vectors of the points P and Q respectively, find (a) The equation of the plane passing through Q and perpendicular to PQ;(b) The perpendicular distance from the point
(a) Determine the equation of the plane that passes through the points (1, 2, –2), (–1, 1, –9) and (2, –2, –12). Find the perpendicular distance from the origin to this plane.(b) Calculate
Find the point P on the line L through the points A(5, 1, 7) and B(6, 0, 8) and the point Q on the line M through the points C(3, 1, 3) and D(–1, 3, 3) such that the line through P and Q is
The angular momentum vector H of a particle of mass m is defined byshow that if r is perpendicular to ω then H = mr2ω. Given that m = 100, r = 0.1(i + j + k) and ω = 5i + 5j – 10k calculate(a)
A particle of mass m, charge e and moving with velocity n in a magnetic field of strength H is known to have accelerationwhere c is the speed of light. Show that the component of acceleration
A force F is of magnitude 14 N and acts at the point A(3, 2, 4) in the direction of the vector –2i + 6j + 3k. Find the moment of the force about the point B(1, 5, –2). Find also the angle between
Points A, B, C have coordinates (1, 2, 1), (–1, 1, 3) and (–2, –2, –2) respectively. Calculate the vector product AB(vector) × AC(vector), the angle BAC and a unit vector perpendicular to
A plane ∏ passes through the three non-collinear points A, B and C having position vectors a, b and c respectively. Show that the parametric vector equation of the plane ∏ is r = a + λ(b
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