Vectors and matrices › Vectors in the plane

Components and addition

Sun's angle at latitude 42° ●○○○○ 10 ptsA building is located at latitude $42^\circ$ . Using $\text{Angle of sun} = 90^\circ - \text{latitude} + (-23.5^\circ)\cdot\cos\left[(N+10)\cdot\frac{360^\circ}{365}\right],$ where $N$ is the day of the year, find the angle of inclination of the sun on March 10th, the 69th day of the year. Round to two decimal places.
Inclination angle from a direction vector ●○○○○ 10 ptsThe direction vector of the line $l$ has coordinates $(2, 2\sqrt{3})$ . Find the inclination angle of the line $l$ in degrees, where the angle lies in $[0^\circ, 180^\circ)$ .
Cross product of two vectors ●○○○○ 10 ptsFind $(2\hat{i}+3\hat{j}-\hat{k})\times(3\hat{j}+2\hat{k})$ . Write the result as $a\hat{i}+b\hat{j}+c\hat{k}$ and enter the coefficients $a$ , $b$ , $c$ .
Angles of a unit vector with the axes ●●○○○ 20 ptsA unit vector $\vec a$ makes an angle $\frac{\pi}{3}$ with $\hat i$ , an angle $\frac{\pi}{4}$ with $\hat j$ , and an acute angle $\theta$ with $\hat k$ . Find $\theta$ in degrees, and hence find the components of $\vec a$ along $\hat i$ , $\hat j$ , $\hat k$ , in that order. Round the second component to two decimal places.
Direction of a vector in quadrant II ●●○○○ 20 ptsFind the direction of the vector $(-4,3)$ , measured as the counterclockwise angle from the positive $x$ -axis in degrees. Round to two decimal places.
Second directional derivative of xyz ●●○○○ 20 ptsFind the second directional derivative of $p(x,y,z)=xyz$ at $(2,3,1)$ in the direction of the vector $\mathbf{i}-\mathbf{j}-\mathbf{k}$ , using the unit vector $\frac{1}{\sqrt{3}}(1,-1,-1)$ for both directions. Round to two decimal places.

More in Vectors in the plane

Dot product Length of a vector combination Linear combinations and collinear points Minimising dot products Parallel vectors Perpendicular vectors Projections Vectors and triangle centres