Express a vector in Cartesian (component) form using the unit vectors 𝒊̂, 𝒋̂ and 𝒌̂.

A vector \(\mathbf{v}\) has magnitude 6 units and makes equal angles with the positive \(x\)-axis, \(y\)-axis, and \(z\)-axis. (a) Express \(\mathbf{v}\) in Cartesian component form using the unit vectors \(\hat{\mathbf{i}}\), \(\hat{\mathbf{j}}\), and \(\hat{\mathbf{k}}\). (2 marks)

The diagram shows vector \(\vec{OA}\) with initial point at the origin \(O(0, 0, 0)\) and terminal point \(A(3, -2, 5)\). Express \(\vec{OA}\) in Cartesian component form using the unit vectors \(\hat{i}\), \(\hat{j}\) and \(\hat{k}\).

A vector \(\vec{AB}\) has initial point \(A(3, -2, 5)\) and terminal point \(B(7, 1, -3)\). (a) Determine the component form of \(\vec{AB}\). (1 mark) (b) Express \(\vec{AB}\) in terms of the unit vectors \(\hat{i}\), \(\hat{j}\) and \(\hat{k}\). (1 mark) (c) Find the magnitude of \(\vec{AB}\), correct to two decimal places. (1 mark)

A displacement vector \(\vec{OA}\) is defined by the position of point \(A\) relative to the origin \(O\). Another displacement vector \(\vec{OB}\) is defined by the position of point \(B\) relative to the origin. The table below shows the coordinates of points \(A\) and \(B\). a) Express the displacement vector \(\vec{OA}\) in Cartesian component form using the unit vectors \(\hat{\mathbf{i}}\), \(\hat{\mathbf{j}}\) and \(\hat{\mathbf{k}}\). (1 mark) b) Express the displacement vector \(\vec{OB}\) in Cartesian component form using the unit vectors \(\hat{\mathbf{i}}\), \(\hat{\mathbf{j}}\) and \(\hat{\mathbf{k}}\). (1 mark) c) Find the vector \(\vec{AB}\) and express it in Cartesian component form using the unit vectors \(\hat{\mathbf{i}}\), \(\hat{\mathbf{j}}\) and \(\hat{\mathbf{k}}\). (2 marks)

A vector has initial point P(3, -2, 5) and terminal point Q(7, 1, -3). Express the vector PQ in Cartesian form using the unit vectors i, j and k.