1. Vectors

Vectors

Ex 1.

For vectors in space

\[\mathbf{u}=[1,2,-1]\quad\text{and}\quad\mathbf{v}=[2,-1,3]\]

calculate \(\mathbf{u}+\mathbf{v}\), \(\mathbf{u}-\mathbf{v}\), the dot product \(\mathbf{u}\cdot\mathbf{v}\), and the norms \(\|\mathbf{u}\|\) and \(\|\mathbf{v}\|\). Check if the vectors are orthogonal.

Ex 2.

For points \(A(1,0,2)\), \(B(3,-1,1)\), and \(C(2,2,0)\), calculate vectors AB and AC and determine the angle between them.

Ex 3.

Calculate the cross product \(\mathbf{u}\times\mathbf{v}\) for the vectors from Ex 1 and check if it is orthogonal to both vectors.

Ex 4.

For vectors on the plane: \(\mathbf{a}=[3,4]\) and \(\mathbf{b}=[-4,3]\), calculate their dot product and check if they are perpendicular. Determine the projection of vector \(\mathbf{a}\) onto \(\mathbf{b}\).

Ex 5.

Calculate the length of the vector \(\mathbf{c} = [1, 1]\) and find the unit vector of this vector.

Ex 6.

Calculate the length of the vector \(\mathbf{c} = [1, 2, 3]\) and find the unit vector of this vector.

Ex 7.

Calculate the area of the triangle spanned by vectors \([2, 1, 2]\) and \([-1, 1,1]\).

Ex 8.

Calculate the angle in degrees between vectors \([4,2,1]\) and \([1,3,2]\).

Ex 9.

Find the coordinates of the midpoint of the segment with endpoints \(A(-1, 2)\) and \(B(3, -2)\).

Ex 10.

For three-dimensional vectors: \(\mathbf{a}=[a_x, a_y, a_z]\), \(\mathbf{b}=[b_x, b_y, b_z]\), \(\mathbf{c}=[c_x, c_y, c_z]\), prove that the following identity holds:

\[ \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \cdot \mathbf{c}) \mathbf{b} - (\mathbf{a} \cdot \mathbf{b}) \mathbf{c}. \]

Ex 11.

Find the most general form of a vector that is simultaneously perpendicular to

\[\mathbf{v}=[-1,3,0]\qquad and \qquad \mathbf{u}=[0,1,1]\]

Ex 12.

For what values of parameters \(p\) and \(q\) are the vectors \(\mathbf{a}=[1-p,3,-1]\) and \(\mathbf{b}=[-2,4-q,2]\) parallel?

Ex 13.

For what values of parameter \(s\) are the vectors \(\mathbf{p}=[s,2,1-s]\) and \(\mathbf{q}=[s,1,-2]\) perpendicular?

Ex 14.

Prove that two vectors must have equal lengths if their sum is perpendicular to their difference.

Ex 15.

\(\star\) We have 2 people (A and B) walking according to the formulas:

A: \((4,5)+(1,-2) t\)

B: \((1,-8)+(2,4) t\)

where \(t\) denotes time. For what \(t\) will the people be closest to each other?