Vector Fundamentals, Part II
Vector Fundamentals, Part III

1. Given \boldsymbol{u} =\left \langle 5 , -1\right \rangle and \boldsymbol{v} =\left \langle 2 ,4\right \rangle, provide a geometric argument of 2\boldsymbol{u} -\frac{1}{2}\boldsymbol{v} and \boldsymbol{u} +\boldsymbol{v} .

2. Find a vector in xy-plane with length \sqrt{2} and angle 60^{ \circ } with respect to positive side of x-axis.

3. Find a unit vector in opposite direction of \left \langle a ,a^{2} ,1\right \rangle .

4. Suppose a and b are real numbers. Find a nonzero vector perpendicular to the vector \left \langle 1 ,a ,b\right \rangle.

5. Find the vector projection of \left \langle 1 ,1 , -1\right \rangle onto \left \langle 4 ,2 ,0\right \rangle .

6. Find all values of a such that the two vectors \left \langle 1 ,a ,a\right \rangle and \left \langle 2 ,a , -4\right \rangle are perpendicular to one another.

7. Find equation of a line passing the point \left (1 ,0 ,1\right ) and perpendicular to the plane x -y -z =9.

8. Find equation of a line passing the point (a ,b ,c) and parallel to the line x =t ,y =2t +1 ,z =1 -t.

9. Find equation of a line passing the points \left (1 , -1 ,\sqrt{2}\right ) and \left (1 , -1 ,\sqrt{3}\right ).

10. Find the distance between the point \left (1 , -1 ,\sqrt{2}\right ) and the line intersection of the planes x -y -z =1 and x +z =1.

11. Find the distance between the two planes x -y =2 +z and x -y =3 +z .

12. Find the angle between two planes x +y +z =1 and x +z =1.

13. Find if the two lines intersect each other. If so, find the point of intersection.

(1)   \begin{align*}l_{1} :x =2t_{1} -6 ,y =4t_{1} +1 ,z = -3t_{1} +2 \\ l_{2} :x =t_{2} +2 ,y =3t_{2} -1 ,z =2t_{2}\end{align*}

 

(2)   \begin{align*}l_{1} :x =2 +t_{1} ,y =1 -t_{1} ,z =3 +3t_{1} \\ l_{2} :x =2 -\sqrt{2} -t_{2} ,y =1 +2\sqrt{2} +2t_{2} ,z =4 +\sqrt{2} +t_{2}\end{align*}

14. Find equation of a plane containing the points \left (1 , -1 ,1\right ) ,\left (0 ,2 ,3\right ),(2,0,0).

15. Find equation of a plane that contains the line x =t +2 ,y =3t -1 ,z =2t and is parallel to the plane x -y +2z =10.

16. Find a unit vector that is perpendicular to \boldsymbol{i} -\boldsymbol{j} and 2\boldsymbol{k}

17. Find the value of t such that the two planes tx -y +2z =9 and z -x =3 are perpendicular to each other.

To see the notes, visit vector fundamentals part II.

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