Quiz about Exploring Vectors and Complex Numbers

Question 1

What is the absolute value (modulus) of the complex number $$z = 3 + 4i$$ ?

Accepted Answer

$$5$$

Answer

$$3$$

Answer

$$4$$

Answer

$$7$$

Question 2

Which of the following pairs of vectors are orthogonal?

Accepted Answer

$$\vec{a} = (1, 0)$$ and $$\vec{b} = (0, 1)$$

Answer

$$\vec{a} = (1, 2)$$ and $$\vec{b} = (2, 4)$$

Answer

$$\vec{a} = (1, 1)$$ and $$\vec{b} = (1, 1)$$

Answer

$$\vec{a} = (2, 3)$$ and $$\vec{b} = (3, 2)$$

Question 3

The magnitude of the vector $$\vec{v} = (a, b)$$ is:

Accepted Answer

$$\sqrt{a^2 + b^2}$$

Answer

$$a + b$$

Answer

$$a^2 + b^2$$

Answer

$$|a - b|$$

Question 4

If $$z = 1 - i$$ , what is $$|z|$$ ?

Accepted Answer

$$\sqrt{2}$$

Answer

$$1$$

Answer

$$2$$

Answer

$$0$$

Question 5

Two vectors $$\vec{a}$$ and $$\vec{b}$$ are orthogonal if:

Accepted Answer

Their dot product is zero

Answer

Their magnitudes are equal

Answer

Their sum is zero

Answer

Their cross product is zero

Question 6

Which of the following is the trigonometric form of the complex number $$z = 2(\cos 60^\circ + i\sin 60^\circ)$$ ?

Accepted Answer

$$z = 2(\cos 60^\circ + i\sin 60^\circ)$$

Answer

$$z = 2 + 2i$$

Answer

$$z = 1 + i\sqrt{3}$$

Answer

$$z = 2e^{i60^\circ}$$

Question 7

If $$\vec{a} = (2, -3)$$ and $$\vec{b} = (3, 2)$$ , what is their dot product?

Accepted Answer

$$-5$$

Answer

$$0$$

Answer

$$12$$

Answer

$$-12$$

Question 8

The absolute value of a complex number $$z = a + bi$$ is given by:

Accepted Answer

$$\sqrt{a^2 + b^2}$$

Answer

$$a + b$$

Answer

$$a^2 + b^2$$

Answer

$$|a - b|$$

Question 9

Which of the following complex numbers has an absolute value of $$1$$ ?

Accepted Answer

All of the above

Answer

$$1 + 0i$$

Answer

$$0 + i$$

Answer

$$-\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}}i$$

Question 10

If two vectors are orthogonal, the angle between them is:

Accepted Answer

$$90^\circ$$

Answer

$$0^\circ$$

Answer

$$45^\circ$$

Answer

$$180^\circ$$

Question 11

What is the magnitude of the vector $$\vec{v} = (6, 8)$$ ?

Accepted Answer

$$10$$

Answer

$$14$$

Answer

$$48$$

Answer

$$2$$

Question 12

Which of the following statements is true about orthogonal vectors?

Accepted Answer

They form a right angle

Answer

They are parallel

Answer

Their dot product is $$1$$

Answer

Their sum is zero

Question 13

The trigonometric form of $$z = 1 + i$$ is:

Accepted Answer

$$\sqrt{2}(\cos 45^\circ + i\sin 45^\circ)$$

Answer

$$2(\cos 30^\circ + i\sin 30^\circ)$$

Answer

$$1(\cos 90^\circ + i\sin 90^\circ)$$

Answer

$$2(\cos 60^\circ + i\sin 60^\circ)$$

Question 14

According to De Moivre's Theorem, $$(\cos heta + i\sin heta)^n$$ equals:

Accepted Answer

$$\cos n heta + i\sin n heta$$

Answer

$$n(\cos heta + i\sin heta)$$

Answer

$$\cos heta^n + i\sin heta^n$$

Answer

$$n\cos heta + i n\sin heta$$

Question 15

If $$z = 5(\cos 0^\circ + i\sin 0^\circ)$$ , what is $$|z|$$ ?

Accepted Answer

$$5$$

Answer

$$0$$

Answer

$$1$$

Answer

$$10$$

Question 16

Which of the following is NOT a property of the absolute value of a complex number?

Accepted Answer

$$|z_1 z_2| = |z_1| + |z_2|$$

Answer

$$|z| \geq 0$$

Answer

$$|z| = 0$$ if and only if $$z = 0$$

Answer

$$|z_1 z_2| = |z_1||z_2|$$

Question 17

If $$\vec{a} = (x_1, y_1)$$ and $$\vec{b} = (x_2, y_2)$$ , their dot product is:

Accepted Answer

$$x_1 x_2 + y_1 y_2$$

Answer

$$x_1 y_2 + x_2 y_1$$

Answer

$$x_1 x_2 - y_1 y_2$$

Answer

$$x_1 y_1 + x_2 y_2$$

Question 18

Which of the following vectors is orthogonal to $$\vec{a} = (4, 3)$$ ?

Accepted Answer

$$(-3, 4)$$

Answer

$$(3, 4)$$

Answer

$$(4, -3)$$

Answer

$$(1, 1)$$

Question 19

The absolute value of $$z = -7i$$ is:

Accepted Answer

$$7$$

Answer

$$-7$$

Answer

$$0$$

Answer

$$1$$

Question 20

If $$z = 2 + 2i$$ , what is $$|z|$$ ?

Accepted Answer

$$2\sqrt{2}$$

Answer

$$2$$

Answer

$$4$$

Answer

$$8$$

Question 21

If $$\vec{a} = (1, 2)$$ and $$\vec{b} = (3, 4)$$ , what is $$\vec{a} + \vec{b}$$ ?

Accepted Answer

$$(4, 6)$$

Answer

$$(2, 2)$$

Answer

$$(2, 6)$$

Answer

$$(3, 8)$$

Question 22

What is the scalar multiple $$3\vec{v}$$ if $$\vec{v} = (2, -1)$$ ?

Accepted Answer

$$(6, -3)$$

Answer

$$(2, -1)$$

Answer

$$(3, -1)$$

Answer

$$(6, 1)$$

Question 23

If $$\vec{a} = (5, 0)$$ and $$\vec{b} = (0, 12)$$ , what is the magnitude of $$\vec{a} + \vec{b}$$ ?

Accepted Answer

$$13$$

Answer

$$12$$

Answer

$$5$$

Answer

$$17$$

Question 24

Given $$\vec{a} = (4, -2)$$ and $$\vec{b} = (-1, 3)$$ , what is $$\vec{a} - \vec{b}$$ ?

Accepted Answer

$$(5, -5)$$

Answer

$$(3, -5)$$

Answer

$$(5, 1)$$

Answer

$$(3, 1)$$

Question 25

If $$\vec{u} = (x, y)$$ , which of the following is the zero vector?

Accepted Answer

$$(0, 0)$$

Answer

$$(x, y)$$

Answer

$$(1, 0)$$

Answer

$$(0, 1)$$

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