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mathematics
precalculus

Questions and Answers of Precalculus

Prove the property of cross products (Theorem 11).Property 1: a x b = -b x a
Show that (a x b) • b = 0 for all vectors a and b in V3.
Show that 0 x a = 0 = a x 0 for any vector a in V3.
Find two unit vectors orthogonal to both j - k and i + j.
Find two unit vectors orthogonal to both (3, 2, 1) and (-1, 1, 0).
If a = (1, 0, 1), b = (2, 1, -1), and c = (0, 1, 3), show that a x (b x c) ≠ (a x b) x c.
If a = (2, -1, 3) and b = (4, 2, 1), find a x b and b x a.
The figure shows a vector a in the xy-plane and a vector b in the direction of k. Their lengths are |a| = 3 and |b| = 2.(a) Find |a x b|.(b) Use the right-hand rule to decide whether the components
Find |u x v | and determine whether u x v is directed into the page or out of the page. |v|= 16 120° l미%3D 12
Find |u x v | and determine whether u x v is directed into the page or out of the page. |v|= 5 45° l미%3 4
Find the vector, not with determinants, but by using properties of cross products.(i + j) x (i - j)
Find the vector, not with determinants, but by using properties of cross products.(j - k) x (k - i)
Find the vector, not with determinants, but by using properties of cross products.k x (i - 2j)
Find the vector, not with determinants, but by using properties of cross products.(i x j) x k
If a = i - 2k and b = j + k, find a x b. Sketch a, b, and a x b as vectors starting at the origin.
Find the cross product a x b and verify that it is orthogonal to both a and b.a = (t, 1, 1/t), b = (t2, t2, 1)
Find the cross product a x b and verify that it is orthogonal to both a and b.a = t i + cos t j + sin tk, b = i - sin t j + cos tk
Find the cross product a x b and verify that it is orthogonal to both a and b.a = 1/2 i + 1/3 j + 1/4 k, b = i + 2 j - 3k
Find the cross product a x b and verify that it is orthogonal to both a and b.a = 3i + 3j - 3k, b = 3i - 3j + 3k
Find the cross product a x b and verify that it is orthogonal to both a and b.a = 2j - 4k, b = -i + 3j + k
Find the cross product a x b and verify that it is orthogonal to both a and b.a = (4, 3, -2), b = (2, -1, 1)
Find the cross product a x b and verify that it is orthogonal to both a and b.a = (2, 3, 0), b = (1, 0, 5)
If θ is the angle between vectors a and b, show that proj, b · proj, a = (a · b) cos²0
Show that if u + v and u - v are orthogonal, then the vectors u and v must have the same length.
The Parallelogram Law states that(a) Give a geometric interpretation of the Parallelogram Law.(b) Prove the Parallelogram Law. (See the hint in Exercise 62.) | a + b ² + | a – b |² = 2|a| + 2|b|P
The Triangle Inequality for vectors is |a + b | < |a| + |b|(a) Give a geometric interpretation of the Triangle Inequality.(b) Use the Cauchy-Schwarz Inequality from Exercise 61 to prove the
Use Theorem 3 to prove the Cauchy-Schwarz Inequality:|a • b | < |a ||b |
Suppose that all sides of a quadrilateral are equal in length and opposite sides are parallel. Use vector methods to show that the diagonals are perpendicular.
Prove Properties 2, 4, and 5 of the dot product (Theorem 2).
If c = |a |b + |b|a, where a, b, and c are all nonzero vectors, show that c bisects the angle between a and b.
A molecule of methane, CH4, is structured with the four hydrogen atoms at the vertices of a regular tetrahedron and the carbon atom at the centroid. The bond angle is the angle formed by the
Find the angle between a diagonal of a cube and a diagonal of one of its faces.
Find the angle between a diagonal of a cube and one of its edges.
If r = (x, y, z), a = (a1, a2, a3), and b = (b1, b2, b3), show that the vector equation (r - a) • (r - b) = 0 represents a sphere, and find its center and radius.
Use a scalar projection to show that the distance from a point P1(x1, y1) to the line ax + by + c = 0 isUse this formula to find the distance from the point (-2, 3) to the line 3x - 4y + 5 = 0. |axı
A boat sails south with the help of a wind blowing in the direction S36°E with magnitude 400 lb. Find the work done by the wind as the boat moves 120 ft.
A sled is pulled along a level path through snow by a rope.A 30-lb force acting at an angle of 40° above the horizontal moves the sled 80 ft. Find the work done by the force.
A tow truck drags a stalled car along a road. The chain makes an angle of 30° with the road and the tension in the chain is 1500 N. How much work is done by the truck in pulling the car 1 km?
Find the work done by a force F = 8 i - 6 j + 9k that moves an object from the point (0, 10, 8) to the point (6, 12, 20) along a straight line. The distance is measured in meters and the force in
Suppose that a and b are nonzero vectors.(a) Under what circumstances is compa b = compb a?(b) Under what circumstances is proja b = projb a?
If a = (3, 0, -1), find a vector b such that compa b = 2.
For the vectors in Exercise 40, find ortha b and illustrate by drawing the vectors a, b, proja b, and ortha b.
Show that the vector ortha b = b - proja b is orthogonal to a. (It is called an orthogonal projection of b.)
Find the scalar and vector projections of b onto a.a = i + 2j + 3k, b = 5i - k
Find the scalar and vector projections of b onto a.a = 3i - 3j + k, b = 2i + 4j - k
Find the scalar and vector projections of b onto a.a = (-1, 4, 8), b = (12, 1, 2)
Find the scalar and vector projections of b onto a.a = (4, 7, -4), b = (3, -1, 1)
Find the scalar and vector projections of b onto a.a = (1, 4), b = (2, 3)
Find the scalar and vector projections of b onto a.a = (-5, 12), b = (4, 6)
If a vector has direction angles a = π/4 and B = π/3, find the third direction angle y.
Find the direction cosines and direction angles of the vector. (Give the direction angles correct to the nearest degree.)(c, c, c), where c > 0
Find the direction cosines and direction angles of the vector. (Give the direction angles correct to the nearest degree.)1/2i + j + k
Find the direction cosines and direction angles of the vector. (Give the direction angles correct to the nearest degree.)i - 2j - 3k
Find the direction cosines and direction angles of the vector. (Give the direction angles correct to the nearest degree.)(6, 3, -2)
Find the direction cosines and direction angles of the vector. (Give the direction angles correct to the nearest degree.)(2, 1, 2)
Find the acute angles between the curves at their points of intersection. (The angle between two curves is the angle between their tangent lines at the point of intersection.)y = sin x, y = cos x, 0
Find the acute angles between the curves at their points of intersection. (The angle between two curves is the angle between their tangent lines at the point of intersection.)y = x2, y = x3
Find the acute angle between the lines.x + 2y = 7, 5x - y = 2
Find the acute angle between the lines.2x - y = 3, 3x + y = 7
Find two unit vectors that make an angle of 608 with v = (3, 4).
Find a unit vector that is orthogonal to both i + j and i + k.
Find the values of x such that the angle between the vectors (2, 1, -1), and (1, x, 0) is 45o.
Use vectors to decide whether the triangle with vertices P(1, -3, -2), Q(2, 0, -4), and R(6, -2, -5) is right-angled.
Find, correct to the nearest degree, the three angles of the triangle with the given vertices.A(1, 0, -1), B(3, -2, 0), C(1, 3, 3)
Find, correct to the nearest degree, the three angles of the triangle with the given vertices.P(2, 0), Q(0, 3), R(3, 4)
Find the angle between the vectors. (First find an exact expression and then approximate to the nearest degree.)a = 8i - j + 4k, b = 4j + 2k
Find the angle between the vectors. (First find an exact expression and then approximate to the nearest degree.)a = 4i - 3j + k, b = 2i - k
Find the angle between the vectors. (First find an exact expression and then approximate to the nearest degree.)a = (- 1, 3, 4), b = (5, 2, 1)
Find the angle between the vectors. (First find an exact expression and then approximate to the nearest degree.)a = (1, -4, 1), b = (0, 2, -2)
Find the angle between the vectors. (First find an exact expression and then approximate to the nearest degree.)a = (-2, 5), b = (5, 12)
Find the angle between the vectors. (First find an exact expression and then approximate to the nearest degree.)a = (4, 3), b = (2, -1)
A street vendor sells a hamburgers, b hot dogs, and c soft drinks on a given day. He charges $4 for a hamburger, $2.50 for a hot dog, and $1 for a soft drink. If A = (a, b, c) and P = (4, 2.5, 1),
(a) Show that i • j = j • k = k • i = 0.(b) Show that i • i = j • j = k • k = 1.
If u is a unit vector, find u • v and u • w. u V
If u is a unit vector, find u • v and u • w.
Find a • b.lal = 80, IbI =50, the angle between a and b is 3π/4
Find a • b.lal= 7, Ibl = 4, the angle between a and b is 30°
Find a • b.a = 3i + 2j - k, b = 4i + 5k
Find a • b.a= 2i + j, b = 1-j+k
Find a • b.a = (p, -p, 2p), b = (2q, q, -q)
Find a • b.a = (4, 1, 1/4), b = (6, -3, -8)
Find a • b.a = (6, -2,3), b = (2, 5, -1)
Find a • b.a = (1.5, 0.4), b = (-4, 6)
Find a • b.a = (5, -2), b = (3, 4)
Suppose the three coordinate planes are all mirrored and a light ray given by the vector first strikes the xz-plane, as shown in the figure. Use the fact that the angle of incidence equals
Use vectors to prove that the line joining the midpoints of two sides of a triangle is parallel to the third side and half its length.
Prove Property 5 of vectors algebraically for the case n = 3. Then use similar triangles to give a geometric proof.
Figure 16 gives a geometric demonstration of Property 2 of vectors. Use components to give an algebraic proof of this fact for the case n = 2.
Suppose that a and b are nonzero vectors that are not parallel and c is any vector in the plane determined by a and b. Give a geometric argument to show that c can be written as c = sa + tb for
(a) Draw the vectors(b) Show, by means of a sketch, that there are scalars s and t such that c = sa + tb.(c) Use the sketch to estimate the values of s and t.(d) Find the exact values of s and t. a =
If A, B, and C are the vertices of a triangle, find CÁ AB + BC + CẢ
Find the unit vectors that are parallel to the tangent line to the parabola y = x2 at the point (2, 4).
Three forces act on an object. Two of the forces are at an angle of 100° to each other and have magnitudes 25 N and 12 N. The third is perpendicular to the plane of these two forces and has
A boatman wants to cross a canal that is 3 km wide and wants to land at a point 2 km upstream from his starting point. The current in the canal flows at 3.5 km/h and the speed of his boat is 13
The tension T at each end of a chain has magnitude 25 N (see the figure). What is the weight of the chain? 37° 37°
A block-and-tackle pulley hoist is suspended in a warehouse by ropes of lengths 2 m and 3 m. The hoist weighs 350 N. The ropes, fastened at different heights, make angles of 50° and 38° with the
A crane suspends a 500-lb steel beam horizontally by support cables (with negligible weight) attached from a hook to each end of the beam. The support cables each make an angle of 60° with the beam.
A woman walks due west on the deck of a ship at 3 mi/h. The ship is moving north at a speed of 22 mi/h. Find the speed and direction of the woman relative to the surface of the water.
The magnitude of a velocity vector is called speed. Suppose that a wind is blowing from the direction N45°W at a speed of 50 km/h. (This means that the direction from which the wind blows is 45°
Find the magnitude of the resultant force and the angle it makes with the positive x-axis. Ул 200 N 300 N 60° х

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