Vector Algebra

Study vectors given by their coordinates in space, including linear operations, length and direction, and the three products (scalar, vector, and mixed) used to find angles, areas, volumes, and normals.

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BASIC DEFINITIONS
Basic Definitions
The core operations on vectors given by coordinates — sums and length, the scalar, vector and mixed products, and projection onto an axis — the tools used throughout the problems below.
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1Problem 1 · Angle Between an Edge and a Space DiagonalConsider the parallelepiped ABCDA′B′C′D′ABCDA'B'C'D' with edge vectors AB→=(6,2,2)\overrightarrow{AB}=(6,2,2), AD→=(1,1,−1)\overrightarrow{AD}=(1,1,-1), and AA′→=(−2,3,1)\overrightarrow{AA'}=(-2,3,1). Compute the angle φ\varphi between edge A′D′A'D' and space diagonal DB′DB'.Start →2Problem 2 · Area of a Triangular SectionConsider the parallelepiped ABCDA′B′C′D′ABCDA'B'C'D' with edge vectors AB→=(6,2,2)\overrightarrow{AB}=(6,2,2), AD→=(1,1,−1)\overrightarrow{AD}=(1,1,-1), and AA′→=(−2,3,1)\overrightarrow{AA'}=(-2,3,1). Compute the area SS of the triangular cross-section EB′CEB'C, where EE is the midpoint of edge D′C′D'C'.3Problem 3 · Volume of a Tetrahedron Inside a ParallelepipedConsider the parallelepiped ABCDA′B′C′D′ABCDA'B'C'D' with edge vectors AB→=(6,2,2)\overrightarrow{AB}=(6,2,2), AD→=(1,1,−1)\overrightarrow{AD}=(1,1,-1), and AA′→=(−2,3,1)\overrightarrow{AA'}=(-2,3,1). Compute the volume of the tetrahedron EB′C′CEB'C'C, where EE is the midpoint of edge D′C′D'C'.4Problem 4 · Unit Normals to a PlaneConsider the parallelepiped ABCDA′B′C′D′ABCDA'B'C'D' with edge vectors AB→=(6,2,2)\overrightarrow{AB}=(6,2,2), AD→=(1,1,−1)\overrightarrow{AD}=(1,1,-1), and AA′→=(−2,3,1)\overrightarrow{AA'}=(-2,3,1). Find the coordinates of both unit vectors n0\mathbf{n}_0 perpendicular to the plane B′CEB'CE, where EE is the midpoint of edge D′C′D'C'.5Problem 5 · Foot of a PerpendicularConsider the parallelepiped ABCDA′B′C′D′ABCDA'B'C'D' with edge vectors AB→=(6,2,2)\overrightarrow{AB}=(6,2,2), AD→=(1,1,−1)\overrightarrow{AD}=(1,1,-1), and AA′→=(−2,3,1)\overrightarrow{AA'}=(-2,3,1). Find the coordinates of the vector AF→\overrightarrow{AF}, where FF is the foot of the perpendicular dropped from point A′A' onto the line ABAB.