Algebraic Vectors

Question 1

What is R² and what does a point in it look like?

Answer 1

R² is a 2D plane. A point is written as P(a, b).

Question 2

What is R³ and what does a point in it look like?

Answer 2

R³ is 3D space. A point is written as P(a, b, c).

Question 3

What are the unit vectors i, j, and k in 2D and 3D?

Answer 3

In 2D: i = (1, 0), j = (0, 1)
In 3D: i = (1, 0, 0), j = (0, 1, 0), k = (0, 0, 1)
Each has a magnitude of 1 and points along its respective axis

Question 4

What does the vector u = (|F|cosθ, |F|sinθ) represent?

Answer 4

It’s a vector expressed using its magnitude |F| and angle θ — the x-component is |F|cosθ (horizontal) and the y-component is |F|sinθ (vertical).

Question 5

What is the algebraic (unit vector) form of vector OP in 2D?

Answer 5

OP = ai + bj, where a and b are the components along the x and y axes.

Question 6

What is the component form of vector OP in 2D and 3D?

Answer 6

2D: OP = (a, b). 3
D: OP = (a, b, c), which can also be written as a column vector (a over b over c).

Question 7

How do you convert from unit vector form to component form? e.g. v = 3i − 2j + 4k

Answer 7

The coefficients of i, j, k become the components — so v = (3, −2, 4).

Question 8

How do you plot a point given a vector in component form? e.g. u = (2, 3, 4)

Answer 8

The components are the coordinates — plot the point (2, 3, 4) by going 2 along x, 3 along y, and 4 along z.

Question 9

How do you plot a point given a vector in unit vector form? e.g. v = 3i − 2j + 4k

Answer 9

Convert to component form first: v = (3, −2, 4), then plot the point (3, −2, 4).

Question 10

How do you add two vectors in component form? e.g. u = (a, b, c) and v = (d, e, f)

Answer 10

Add corresponding components: u + v = (a+d, b+e, c+f).

Question 11

How do you subtract two vectors in component form? e.g. u = (a, b, c) and v = (d, e, f)

Answer 11

ubtract corresponding components: u − v = (a−d, b−e, c−f)

Question 12

How do you multiply a vector by a scalar in component form? e.g. ku = k(a, b, c)

Answer 12

Multiply every component by the scalar: ku = (ka, kb, kc).

Question 13

How do you find the magnitude of a vector in component form? e.g. u = (a, b, c)

Answer 13

In 3D: |u| = √(a² + b² + c²).
In 2D: |u| = √(a² + b²).

Question 14

What does it mean for two vectors to be collinear?

Answer 14

Two vectors are collinear if one is a scalar multiple of the other — they point in the same (or opposite) direction.

Question 15

How do you find a and b if vectors u and v are collinear?

Answer 15

Set u = kv for some scalar k. Match corresponding components to form equations, then solve for a, b, and k.

Example: If u = (2, a) and v = (b, 6) are collinear → 2/b = a/6 = k, solve the system.

Question 16

When drawing a 3D coordinate system, which direction should each axis point?

Answer 16

X → toward you (out of the page), Y → horizontal (right is positive), Z → vertical (up is positive).

Question 17

What is the positive direction for each axis in a standard 3D diagram?

Answer 17

X: toward you. Y: to the right. Z: upward.

Question 18

How do you find the vector between two points P1(x1, y1) and P2(x2, y2) in 2D?

Answer 18

Subtract the coordinates of P1 from P2: P1P2 = (x2−x1, y2−y1). Always go “end minus start.”

Question 19

How do you find the vector between two points P1(x1, y1, z1) and P2(x2, y2, z2) in 3D?

Answer 19

P1P2 = (x2−x1, y2−y1, z2−z1).
Same rule — end minus start for each component.

Question 20

How do you find a unit vector in the same direction as vector u?

Answer 20

Find magnitude of u. Divide u by its magnitude: û = u / |u|. This scales the vector to length 1 while keeping its direction.

Question 21

How do you find a unit vector in the opposite direction to vector v?

Answer 21

Find magnitude of v. Negate v first, then divide by its magnitude: û = −v / |v|. Negating flips the direction, dividing by |v| scales it to length 1.

Question 22

How do you verify that a unit vector is correct

Answer 22

Calculate its magnitude — it must equal 1. e.g. if û = (a, b, c), check that √(a²+b²+c²) = 1.

Question 23

Worked example — find the unit vector in the opposite direction to v = (−2, 2, 5)

Answer 23

|v| = √(4+4+25) = √33.

−v = (2,−2,−5).

û = (2/√33, −2/√33, −5/√33)
= (2√33/33, −2√33/33, −5√33/33)

Question 24

What is the formula to express a vector in component form given its magnitude |u| and angle θ in standard position?

Answer 24

u = (|u|cosθ, |u|sinθ). The x-component uses cosine, the y-component uses sine.

Question 25

In parallelogram OAPB, O is the origin and A, B are opposite vertices. How do you find the coordinates of P?

Answer 25

Since OAPB is a parallelogram, OP = OA + OB, so P = A + B. Add the coordinates of A and B together.

Question 26

What are direction angles of a vector?

Answer 26

The angles α, β, γ that a vector makes with the positive x, y, and z axes respectively, where 0 < α, β, γ < 180°.

Question 27

Which axis does each direction angle correspond to?

Answer 27

α (alpha) → positive x-axis, β (beta) → positive y-axis, γ (gamma) → positive z-axis.

Question 28

What is the formula for the direction cosines of vector v = (a, b, c)?

Answer 28

cosα = a/|v| cosβ = b/|v| cosγ = c/|v| where |v| = √(a²+b²+c²).

Question 29

How do you find a direction angle from its cosine?

Answer 29

Take the inverse cosine: α = cos⁻¹(a/|v|), and same for β and γ. Since angles are between 0° and 180°, cos⁻¹ always gives the correct angle.

Question 30

What is a useful identity that the direction cosines always satisfy?

Answer 30

cos²α + cos²β + cos²γ = 1. This is because a²+b²+c² = |v|², so dividing both sides by |v|² gives this result.

Question 31

What is the theorem for determining if three vectors are coplanar using the cross product?

Answer 31

If (u × v) · w = 0, then u, v, w are coplanar — they all lie on the same plane. The cross product u × v gives the normal to the plane, and if w dots to 0 with that normal, w also lies on that plane.

Question 32

How do you determine if two vectors are collinear?

Answer 32

Check if one is a scalar multiple of the other — u = kv for some scalar k. If the ratios of all corresponding components are equal, they are collinear.

Question 33

What is a linear combination of vectors?

Answer 33

Writing a vector x as x = au + bv (or au + bv + cw in 3D) for some scalars a and b. You are expressing x as a sum of scaled versions of other vectors.

Question 34

How do you write x = (5, 8) as a linear combination of u = (−1, 3) and v = (2, 4)?

Answer 34

Set x = au + bv: (5, 8) = a(−1, 3) + b(2, 4). This gives the system: −a + 2b = 5 and 3a + 4b = 8. Solving: from first equation a = 2b − 5, substitute: 3(2b−5) + 4b = 8 → 10b = 23 → b = 2.3, a = −0.4. So x = −0.4u + 2.3v.

Question 35

How do you determine if vectors are collinear or coplanar in 3D?

Answer 35

First check collinearity — is one a scalar multiple of another? If not, check coplanarity using the triple scalar product: if (u × v) · w = 0, they are coplanar. If (u × v) · w ≠ 0, they are neither — they span all of 3D space and form a basis.

Question 36

What does it mean for 3 vectors to form a basis for 3D space?

Answer 36

They are non-coplanar (triple scalar product ≠ 0), meaning they span all of R³ — any vector in 3D space can be written as a linear combination of these 3 vectors.