Compute \((2,3)\cdot(4,-1)\).

Find \((1,-2,5)\cdot(3,0,-1)\).

Given \(|\mathbf a|=6\), \(|\mathbf b|=5\), and angle \(60^\circ\), find \(\mathbf a\cdot\mathbf b\).

Test whether \((2,1)\) and \((3,-6)\) are perpendicular.

Find \(|(2,-3,6)|\) using a dot product.

A force \(\mathbf F=(6,2)\) N moves a particle by \(\mathbf s=(3,0)\) m. Find the work done.

Find the angle between \((1,0)\) and \((1,1)\).

Find the scalar projection of \(\mathbf a=(3,4)\) onto \(\mathbf b=(1,0)\).

Find the vector projection of \(\mathbf a=(2,3)\) onto \(\mathbf b=(1,1)\).

Find \(k\) if \((k,2)\cdot(3,-6)=0\).

Compute \((\mathbf i+2\mathbf j-\mathbf k)\cdot(3\mathbf i-\mathbf j+4\mathbf k)\).

Use dot products to decide whether the triangle with sides \((3,4)\) and \((-4,3)\) at one vertex is right-angled.

Find \(\mathbf a\cdot(2\mathbf b-\mathbf c)\) for \(\mathbf a=(1,2)\), \(\mathbf b=(3,0)\), \(\mathbf c=(1,-1)\).

Find the component of \((4,4)\) in the direction of unit vector \((1/\sqrt2,1/\sqrt2)\).

Find the angle between \((1,2,2)\) and \((2,0,1)\).

Find \(x\) if \((x,1,2)\) is orthogonal to \((3,-1,4)\).

A force of magnitude \(10\) N acts at \(30^\circ\) to a displacement of \(4\) m. Find the work.

Show that \((\mathbf a+\mathbf b)\cdot(\mathbf a-\mathbf b)=|\mathbf a|^2-|\mathbf b|^2\), then explain what happens when \(|\mathbf a|=|\mathbf b|\).

Find the work done by \(\mathbf F=(2,-1,3)\) along displacement \(\mathbf s=(4,5,-2)\), then find the component of the force in the direction of the displacement.

Find \(\lambda\) so that \((1,\lambda,2)\) makes a \(60^\circ\) angle with \((1,0,0)\), with \(\lambda\ge0\).