91Ó°ÊÓ

If \(a=\left\langle a_{1}, a_{2}\right\rangle, b=\left\langle b_{1}, b_{2}\right\rangle, c=\left\langle c_{1}, c_{2}\right\rangle,\) and \(m\) and \(n\) are real numbers, prove the stated property. $$\mathbf{a}+\mathbf{0}=\mathbf{a}$$

Angles of a triangular plot A triangular plot of land has sides of lengths 420 feet, 350 feet, and 180 feet. Approximate the smallest angle between the sides.

Represent the complex number geometrically. $$(-3 i)(2-i)$$

If a and b are vectors with the same initial point and angle \(\theta\) between them, prove that $$\|\mathbf{a}-\mathbf{b}\|^{2}=\|\mathbf{a}\|^{2}+\|\mathbf{b}\|^{2}-2\|\mathbf{a}\|\|\mathbf{b}\| \cos \theta$$

Determine \(m\) such that the two vectors are orthogonal. $$9I-16 m J, \quad I+4 m J$$

Represent the complex number geometrically. $$(1+i)^{2}$$

If \(a=\left\langle a_{1}, a_{2}\right\rangle, b=\left\langle b_{1}, b_{2}\right\rangle, c=\left\langle c_{1}, c_{2}\right\rangle,\) and \(m\) and \(n\) are real numbers, prove the stated property. $$\mathbf{a}+(-\mathbf{a})=\mathbf{0}$$

An airplane is flying in the direction \(80^{\circ}\) with an airspeed of \(400 \mathrm{mi} / \mathrm{hr}\). Its ground speed and true course are \(390 \mathrm{mi} / \mathrm{hr}\) and \(90^{\circ},\) respectively. Approximate the direction and speed of the wind.

An airplane flies 165 miles from point \(A\) in the direction \(130^{\circ}\) and then travels in the direction \(245^{\circ}\) for 80 miles. Approximately how far is the airplane from \(A ?\)

Represent the complex number geometrically. $$4(-1+2 i)$$