Vectors Progress Test

Let u = i - 2j - 3k and v = 2i + j - k. The vector resolute of u perpendicular to v is:

•  $-\frac{5}{2}$ (j + k)

• -5(i + j)

• -2i  $-\frac{5}{2}$ j  $-\frac{5}{2}$ k

• i +  $-\frac{1}{2}$ j  $-\frac{1}{2}$ k 

• 2i +  $\frac{1}{2}$ j  $-\frac{1}{2}$ k

A unit vector in the opposite direction to 2i - 2j + k is:

• -2i + 2j - k

•  $\frac{1}{2}$  (-2i + 2j - k)

•  $\frac{1}{3}$  (-2i + 2j - k)

•  $\frac{1}{4}$  (-2i + 2j - k)

•  $\frac{1}{5}$  (-2i + 2j - k)

The vectors a = i + j + 2k, b = -i + 2j - 2k, c = i + mk are linearly dependent, where m is a real constant. The value of m is:

• 
  

  $-\frac{1}{3}$
  

• 0

• $\frac{2}{3}$

• 1

• 2

The vectors a = 2i + j - k, b = mi + nj, and c = -i - 3j + k are linearly dependent. If b is a unit vector, then the values of m and n could be:

• 
  

  $m\ =\ -\frac{1}{\sqrt{5}}$ and $n\ =\ \frac{2}{\sqrt{5}}$
  

• $m\ =\ \frac{1}{\sqrt{5}}$ and $n\ =\ \frac{2}{\sqrt{5}}$

• $m\ =\ -\frac{1}{\sqrt{5}}$ and $n\ =\ -\frac{2}{\sqrt{5}}$

• $m\ =\ \frac{1}{\sqrt{3}}$ and $n\ =\ -\frac{2}{\sqrt{3}}$

• $m\ =\ 2\$ and $n\ =\ 1$

If the vectors a = mi -  $\sqrt{m}$  j - 3k and b = mi +  $\sqrt{m}$  j + 2k are perpendicular, then:

• m = 0

• m = 3 and m = -2

• m = -3 and m = 2

• m = 3

• m = -2

Let a = i + j, b = j + k, and c = 2i - j - 3k. Which of the following is false?

• The vectors a and b have the same length

• The angle between vectors a and b is  $60\degree$ 

• The vector a + c is parallel to the vector a - b

• c = 2a - 3b

• The vectors a, b, and c are linearly independent

A, B, and C are three points in space. To prove that ABC is a right-angled isosceles triangle, it is necessary to show:

• $\left|\overline{AB}\right|\left|\overline{AC}\right|\ =\ \overline{AB}\ .\ \overline{AC}$ and $\left|\overline{AB}\right|\ =\ \left|\overline{BC}\right|$

• $\left|\overline{AB}\right|\left|\overline{AC}\right|\ =\ \sqrt{2}\overline{AB}\ .\ \overline{AC}$ and $\left|\overline{AB}\right|\ =\ \left|\overline{BC}\right|$

• $\left|\overline{AB}\right|\ =\ \left|\overline{BC}\right|$ and $\overline{AB}\ .\ \overline{AC}\ =\ 0$

• $\overline{AB}+\overline{BC}+\overline{CA}\ =\ \overrightarrow{0}$ and $\overline{BA}\ .\ \overline{BC}\ =\ 0$

• $\overline{AB}\ +\ \overline{BC}\ +\ \overline{CA}\ =\ \overrightarrow{0}$ and $\left|\overline{AB}\right|\ =\ \left|\overline{BC}\right|$

Two vectors a and b are such that a.b = 0, a.a = 1, and b.b = 2. Which of the following statements is true?

• a is a unit vector and |b - a| =  $\sqrt{3}$  

• a is perpendicular to the vector b and the length of the vector b is 2

• a is perpendicular to the vector b and |b - a| = 3

• a is parallel to the vector b and |b - a| =  $\sqrt{3}$  

• a is parallel to the vector b and the length of the vector b is  $\sqrt{2}$  

Two vectors u and v are such that |u| = |v| = L.  The angle between the vectors u and v is  $\theta$  .  Then, |u - v| is equal to:

• 0

• 2L

•  $\sqrt{2}$  L

• L $\sin\left(\theta\right)$  

• 2L $\sin\left(\frac{\theta}{2}\right)$  

Given the vectors a = i - j + k, b = 2i + 2j - k, and c =  $\frac{1}{3}$  (i - 5j + 4k), then:

• The vectors a, b, and c are linearly dependent

• The vector c is the scalar resolute of the vector a parallel to the vector b

• The vector c is the vector resolute of the vector a parallel to the vector b

• The vector c is the vector resolute of the vector a perpendicular to the vector b

• The vector c is parallel to the vector a + b