Vector Calculus for Engineers Quiz

What are vectors defined as in three-dimensional Euclidean space?

• Line segments with both length and direction

• Only line segments with length

• Only line segments with direction

• Points in space

What is the equation that represents a mass acted on by two forces?

• m = F1 + F2

• ma = F1 + F2

• F = ma

• F1 = F2

What does scalar multiplication of a vector by a positive scalar change?

• The direction of the vector

• The length of the vector

• The magnitude of the vector

• The position of the vector

What is the negative of a vector?

• Reverses its direction

• Increases its magnitude

• Decreases its magnitude

• Makes it a scalar

What are examples of scalars?

• Velocity and acceleration

• Mass and charge

• Force and energy

• Length and area

Show graphically that vector addition is associative, i.e.,(퐴+퐵) +퐶=퐴+ (퐵+퐶).

Using vectors, prove that the line segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length.

Given a Cartesian coordinate system with standard unit vectors푖,푗, and푘, let a mass m1 be at position r1=x1푖+y1푗+z1푘 and a mass m2 be at position r2=x2푖+y2푗+z2푘. In terms of the standard unit vectors, determine the unit vector that points from m1 to m2.

Newton’s law of universal gravitation states that two point masses attract each other along the line connecting them, with a force proportional to the product of their masses and inversely proportional to the square of the distance between them. The magnitude of the force acting on each mass is therefore F=G m1 m2 r2, where m1 and m2 are the two masses, r is the distance between them, and G is the gravitational constant. Let the masses m1 and m2 be located at the position vectors r1 and r2. Write down the vector form for the force acting on m1 due to its gravitational attraction to m2.

Using the definition of the dot product A·B=A1B1+A2B2+A3B3, prove that a) A·B=B·A; b) A·(B+C) =A·B+A·C; c) A·(kB) = (kA)·B=k(A·B).