Direction cosines and direction ratios: Question bank

Before we start, one reminder of the two words we lean on the whole page:

Direction cosines (DCs) $(l, m, n)$ = the three components of a unit vector along the line, so $l^{2} + m^{2} + n^{2} = 1$ .
Direction ratios (DRs) $(a, b, c)$ = any triple proportional to those, no length rule.

True or false — justify

Are direction cosines unique for a given line?

False — a line has two sets of DCs,

(l, m, n)

and

(- l, - m, - n)

, one for each of the two opposite directions you can travel along it.

(3, 3, 3)

a valid set of direction cosines?

False —

3^{2} + 3^{2} + 3^{2} = 27 \neq = 1

, so these are only direction ratios; the matching DCs are

(1/ 3, 1/ 3, 1/ 3)

True or false: any three numbers can be direction ratios of some line.

False — the triple must be not all zero;

(0, 0, 0)

picks out no direction at all.

True or false: if two lines have proportional DRs they are the same line.

False — they are only parallel; two distinct parallel rails have identical direction but sit at different places.

Is it true that

cos^{2} α + cos^{2} β + cos^{2} γ = 1

but

sin^{2} α + sin^{2} β + sin^{2} γ = 2

True — subtract the first identity from

3

\sum (1 - cos^{2}) = 3 - 1 = 2

, so the three direction sines square-sum to

2

, not

1

True or false: a line can make an angle of

3 0^{\circ}

with all three axes.

False — that needs

3 cos^{2} 3 0^{\circ} = 3 \cdot \frac{3}{4} = \frac{9}{4} \neq = 1

, violating

l^{2} + m^{2} + n^{2} = 1

; the equal-angle line uses

cos^{- 1} (1/ 3) \approx 54. 7^{\circ}

True or false: direction angles

α, β, γ

can each independently be any value in

[0, 18 0^{\circ}]

False — they are coupled by

cos^{2} α + cos^{2} β + cos^{2} γ = 1

; fixing two of them constrains the third.

Is the angle formula

cos θ = l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2}

valid only for unit-vector DCs?

True — it assumes both triples have length

1

; for raw DRs you must divide by the product of the two magnitudes first.

Spot the error

"DRs

(2, - 4, 4)

give DCs

(2, - 4, 4) / 2^{2} + 4^{2} + 4^{2} = (2, - 4, 4) /6

." Where's the slip?

The arithmetic is fine —

4 + 16 + 16 = 6

— but the answer should carry a ==

\pm

==; DCs are

\pm (1/3, - 2/3, 2/3)

since either direction along the line is legal.

"A line makes

α = β = γ

, so each angle is

4 5^{\circ}

because

3 \times 45 = 135

... close enough." What's wrong?

The angles don't add to anything meaningful; the real constraint is

3 cos^{2} α = 1

, giving

α = cos^{- 1} (1/ 3) \approx 54. 7^{\circ}

, not

4 5^{\circ}

"To get the angle between DRs

(1, 0, 0)

and

(0, 1, 0)

I compute

1 \cdot 0 + 0 \cdot 1 + 0 \cdot 0 = 0

, so they're parallel." Fix it.

Dot product

= 0

means perpendicular, not parallel; the

x

- and

y

-axes meet at

9 0^{\circ}

"Since

l : m : n = a : b : c

, I can write

l = a

." Why is this false in general?

Proportional is not equal —

l = k a

with

k = \pm 1/ a^{2} + b^{2} + c^{2}

; only if

(a, b, c)

already has length

1

does

k = \pm 1

"The line through

A (1, 1, 1)

and

B (1, 1, 1)

has DRs

(0, 0, 0)

." What went wrong conceptually?

A = B

is a single point, so there is no line and no direction;

(0, 0, 0)

is forbidden precisely because it encodes "no direction".

"Two lines with DRs

(1, 2, 3)

and

(2, 4, 7)

are parallel since the first two ratios match." Spot the error.

Parallelism needs all three ratios equal;

3/7 \neq = 1/2

, so they are not parallel.

cos θ

came out

1.2

, and the lines are steeply tilted." What must be wrong?

A cosine can never exceed

1

; a value

> 1

means you forgot to divide by the magnitudes (or an arithmetic slip) — the DR dot product was not normalised.

Why questions

Why cosines of the angles, and not the angles

α, β, γ

themselves?

Cosines are the unit vector's components, which add and dot like vectors; raw angles have no such algebra and can't be plugged into a dot product.

Why does every line have exactly two sets of DCs but infinitely many DRs?

A unit vector has only two orientations along the line (

\overset{u}{^}

and

- \overset{u}{^}

), fixing DCs to

\pm (l, m, n)

; DRs allow any nonzero scale factor, an infinite family.

Why must the DRs be "not all zero"?

You divide by

a^{2} + b^{2} + c^{2}

to normalise; if all are zero that length is

0

and the division is undefined, matching the fact that no direction exists.

Why is

cos θ = l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2}

just a dot product?

Because each DC triple is a unit vector, and

\overset{u}{^}_{1} \cdot \overset{u}{^}_{2} = ∣ \overset{u}{^}_{1} ∣∣ \overset{u}{^}_{2} ∣ cos θ = cos θ

when both lengths are

1

Why do we usually take the acute angle between two lines?

A line has no arrowhead, so both

θ

and

18 0^{\circ} - θ

describe the crossing; we report the acute one by using

∣ l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2} ∣

Why does the perpendicularity test drop the denominators entirely?

cos 9 0^{\circ} = 0

forces the numerator alone to vanish; the positive magnitudes in the denominator never affect whether the fraction equals zero.

Why can't all three direction cosines be positive and large at once?

Their squares must sum to exactly

1

, so making one close to

1

==forces the others toward

0

== — they compete for a fixed budget.

Edge cases

What are the DCs of a line lying along the positive

x

-axis?

(1, 0, 0)

— it makes

0^{\circ}

with

x

(

cos 0 = 1

) and

9 0^{\circ}

with

y

and

z

(

cos 9 0^{\circ} = 0

What are the direction angles of the

z

-axis?

α = β = 9 0^{\circ}, γ = 0^{\circ}

, giving DCs

(0, 0, 1)

— it leans fully toward the ceiling and not at all toward the other walls.

γ = 9 0^{\circ}

, what does that say geometrically?

n = cos 9 0^{\circ} = 0

, so the line is ==perpendicular to the

z

-axis==, i.e. it lies flat in (a plane parallel to) the

x y

-plane.

A line makes

6 0^{\circ}

with

x

and

6 0^{\circ}

with

y

; is

γ

forced?

Almost —

n^{2} = 1 - \frac{1}{4} - \frac{1}{4} = \frac{1}{2}

, so

n = \pm 1/ 2

and

γ = 4 5^{\circ}

13 5^{\circ}

; the ==

\pm

== gives two mirror directions.

Can a direction cosine equal

- 1

Yes — e.g.

(- 1, 0, 0)

is the negative

x

-axis;

α = 18 0^{\circ}

, and

l^{2} + m^{2} + n^{2} = 1

still holds.

Can two of the three direction angles be obtuse simultaneously?

Yes — e.g.

(- \frac{1}{2}, - \frac{1}{2}, \frac{1}{2})

has two negative cosines (two obtuse angles) yet still square-sums to

1

What happens to the angle formula when the two lines coincide in direction?

The DC triples are equal, so

l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2} = l^{2} + m^{2} + n^{2} = 1

, giving

θ = 0^{\circ}

as expected.

Recall One-line self-test

Cover every answer above, run the list top to bottom, and flag any item where you said the right thing for the wrong reason — those are the real gaps.

Connections

Direction cosines and direction ratios — the parent these traps stress-test.
Vectors and dot product — the reason "dot the units" gives the angle.
Equation of a line in 3D — where DRs reappear as the direction vector.
Angle between line and plane — the same $\pm$ and acute-angle care apply.