(1) Two vectors are given: (\vec{a} = 3\vec{i} - 3\vec{j}) and (\vec{b} = -1\vec{i} + 4\vec{j}). Determine the coordinates of vector (\vec{c}) if the following holds:

$$ \vec{c} = \vec{a} + \vec{b} $$

Rješenje zapiši kao uređeni par (x, y).

(2) Calculate the linear combination of vectors. Vectors (\vec{u} = ( 2, -2 )) and (\vec{v} = ( -3, 3 )) are given. Determine vector (\vec{w}) according to the formula:

$$ \vec{w} = 5 \cdot \vec{u} - 4 \cdot \vec{v} $$

(3) Calculate the magnitude (modulus) of the resultant vector (\vec{z} = \vec{a} - \vec{b}), if the coordinates of the vectors are given in the table:

Napomena: Duljina vektora \(\vec{v}(x, y)\) računa se formulom \( |\vec{v}| = \sqrt{x^2 + y^2} \).

(4) Points (A( 4, 6 )) and (B( 9, 9 )) define vector (\vec{AB}). Point (C) has coordinates (( 15, 5 )). Determine the coordinates of vector (\vec{d}), which is the sum of vector (\vec{AB}) and the radius-vector of point (C) ((\vec{r_C})).

$$ \vec{d} = \vec{AB} + \vec{r_C} $$

(5) Solve the vector equation and determine the unknown vector (\vec{x}). Vectors (\vec{m} = ( 12, -10 )) and (\vec{n} = ( 6, 6 )) are given.

$$ 2\vec{x} + \vec{n} = \vec{m} $$

(6) Two forces, (\vec{F_1}) and (\vec{F_2}), act on a body at a right angle (along the x and y axes). Calculate the magnitude of the resultant force (\vec{R} = \vec{F_1} + \vec{F_2}).

(7) Vectors (\vec{p} = (x, 9)) and (\vec{q} = ( 4, -2 )) are given. If the sum of these vectors (\vec{s} = \vec{p} + \vec{q}) is equal to the vector (( 13, y )), determine the unknown values (x) and (y).

(8) A boat crosses a river. Its velocity relative to the water is (\vec{v_c} = ( 5.50, 2.50 )) m/s, while the river flows at a velocity (\vec{v_r} = ( 1, -1.50 )) m/s. What is the actual velocity of the boat relative to the shore ((\vec{v_u} = \vec{v_c} + \vec{v_r}))?

Rezultat izrazi kao vektor u koordinatnom sustavu.

(9) Verify the associative property of vector addition. Calculate the left side of the equality for the given vectors:

\(\vec{a} = (4, 2)\), \(\vec{b} = (-3, 2)\), \(\vec{c} = (8, -1)\)

$$ \vec{S} = (\vec{a} + \vec{b}) - \vec{c} $$

(10) Determine the perimeter of a triangle whose vertices are points (A(0,0)), (B(3, 0)), and (C(0, 4)) using vector addition and subtraction to determine side lengths.

Uputa: Stranice su vektori \(\vec{AB}\), \(\vec{BC}\) i \(\vec{CA}\). Izračunaj njihove duljine i zbroji ih.