Vector Analysis
1.1.1 Show how to find A and B, given A+B and A −B.
1.1.2 The vector A whose magnitude is 1.732 units makes equal angles with the coordinate axes. Find Ax,Ay , and Az.
1.1.3 Calculate the components of a unit vector that lies in the xy-plane and makes equal angles with the positive directions of the x- and y-axes.
1.1.4 The velocity of sailboat A relative to sailboat B, vrel, is defined by the equation vrel = vA − vB, where vA is the velocity of A and vB is the velocity of B. Determine the velocity of A relative to B if vA = 30 km/hr east vB = 40 km/hr north.
ANS. vrel = 50 km/hr, 53.1◦ south of east.
1.1.5 A sailboat sails for 1 hr at 4 km/hr (relative to the water) on a steady compass heading of 40◦ east of north. The sailboat is simultaneously carried along by a current. At the end of the hour the boat is 6.12 km from its starting point. The line from its starting point to its location lies 60◦ east of north. Find the x (easterly) and y (northerly) components of the water’s velocity.
ANS. veast = 2.73 km/hr, vnorth ≈ 0 km/hr.
1.1.6 A vector equation can be reduced to the form A = B. From this show that the one vector equation is equivalent to three scalar equations. Assuming the validity of Newton’s second law, F = ma, as a vector equation, this means that ax depends only on Fx and is independent of Fy and Fz.
1.1.7 The vertices A,B, and C of a triangle are given by the points (−1, 0, 2), (0, 1, 0), and (1,−1, 0), respectively. Find point D so that the figure ABCD forms a plane parallelogram. ANS. (0,−2, 2) or (2, 0,−2).
1.1.8 A triangle is defined by the vertices of three vectors A,B and C that extend from the origin. In terms of A,B, and C show that the vector sum of the successive sides of the triangle (AB +BC +CA) is zero, where the side AB is from A to B, etc. 1.1.9 A sphere of radius a is centered at a point r1.
(a) Write out the algebraic equation for the sphere.
(b) Write out a vector equation for the sphere.
ANS. (a) (x −x1)2 +(y −y1)2 +
1.1.2 The vector A whose magnitude is 1.732 units makes equal angles with the coordinate axes. Find Ax,Ay , and Az.
1.1.3 Calculate the components of a unit vector that lies in the xy-plane and makes equal angles with the positive directions of the x- and y-axes.
1.1.4 The velocity of sailboat A relative to sailboat B, vrel, is defined by the equation vrel = vA − vB, where vA is the velocity of A and vB is the velocity of B. Determine the velocity of A relative to B if vA = 30 km/hr east vB = 40 km/hr north.
ANS. vrel = 50 km/hr, 53.1◦ south of east.
1.1.5 A sailboat sails for 1 hr at 4 km/hr (relative to the water) on a steady compass heading of 40◦ east of north. The sailboat is simultaneously carried along by a current. At the end of the hour the boat is 6.12 km from its starting point. The line from its starting point to its location lies 60◦ east of north. Find the x (easterly) and y (northerly) components of the water’s velocity.
ANS. veast = 2.73 km/hr, vnorth ≈ 0 km/hr.
1.1.6 A vector equation can be reduced to the form A = B. From this show that the one vector equation is equivalent to three scalar equations. Assuming the validity of Newton’s second law, F = ma, as a vector equation, this means that ax depends only on Fx and is independent of Fy and Fz.
1.1.7 The vertices A,B, and C of a triangle are given by the points (−1, 0, 2), (0, 1, 0), and (1,−1, 0), respectively. Find point D so that the figure ABCD forms a plane parallelogram. ANS. (0,−2, 2) or (2, 0,−2).
1.1.8 A triangle is defined by the vertices of three vectors A,B and C that extend from the origin. In terms of A,B, and C show that the vector sum of the successive sides of the triangle (AB +BC +CA) is zero, where the side AB is from A to B, etc. 1.1.9 A sphere of radius a is centered at a point r1.
(a) Write out the algebraic equation for the sphere.
(b) Write out a vector equation for the sphere.
ANS. (a) (x −x1)2 +(y −y1)2 +