Scalar and vector products - Exercises | IB Math AAHL | Perplex

Scalar and vector products

Introduction to and applications of vector and scalar products

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Exercises

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Key Skills

Scalar product

AHL 3.13

The scalar product, also called the dot product, takes two vectors and produces a scalar (a number). For two vectors v=⎝⎛v1v2v3⎠⎞ and w=⎝⎛w1w2w3⎠⎞, the scalar product is calculated as:

v⋅w=v1w1+v2w2+v3w3📖

This operation combines corresponding components of each vector, resulting in a single numerical value.

Angle between vectors

AHL 3.13

The scalar product also has a geometric interpretation involving the angle θ between two vectors:

v⋅w=∣v∣∣w∣cosθ📖

Equivalently, isolating cosθ:

cosθ=∣v∣∣w∣v1w1+v2w2+v3w3📖

The angle θ is measured between the heads of v and w:

Acute vs Obtuse Angles

AHL 3.13

If the scalar product of two vectors is negative, then

cosθ=∣u∣⋅∣v∣u⋅v<0

and thus θ must be an obtuse angle: 90°<θ≤180°.

But since vectors always form an acute AND an obtuse angle, we can find the acute angle by subtracting

180°−θ

whenever θ>90°.

Perpendicular vectors

AHL 3.13

Since v⋅w=∣v∣∣w∣cosθ, if two vectors are perpendicular then cos90°=0⇒ their scalar product is 0.

Properties of the scalar product

AHL 3.13

For any vectors u, v, and w, and scalar k:

u⋅v=v⋅u🚫

u⋅(v+w)=u⋅v+u⋅w🚫

(ku)⋅v=k(u⋅v)=u⋅(kv)🚫

u⋅u=∣u∣2🚫

Vector product

AHL 3.16

The vector product, sometimes called the cross product, of two vectors v and w is given by:

v×w=⎝⎛v2w3−v3w2v3w1−v1w3v1w2−v2w1⎠⎞📖

This vector is perpendicular to both original vectors. Using the right-hand rule, if your index finger points in the direction of v and your middle finger points towards w, your thumb points towards v×w. The vector product thus creates a new vector perpendicular to both original vectors.

Vector areas with cross product

AHL 3.16

The magnitude of the vector product v×w gives the area of the parallelogram formed by vectors v and w:

Area=∣v×w∣📖

Intuitively, this happens because the magnitude combines both vectors’ lengths and how "spread out" they are from each other, capturing exactly the amount of two-dimensional space they span.

Vector product and sin of angle

AHL 3.16

The magnitude of the vector product is connected to the sine of the angle between the vectors by the formula

∣v×w∣=∣v∣∣w∣sinθ📖

Here, θ is the angle between vectors v and w. This relationship holds because the area of the parallelogram formed by the two vectors depends on their lengths and the angle separating them.

Specifically, the area is largest when the vectors are perpendicular (sin90∘=1) and zero when they are parallel (sin0∘=0).

Components of vectors

AHL 3.16

For two vectors a and b, the magnitude of the component of vector a that acts in the direction of vector b is given by

∣a∣cosθ=∣b∣a⋅b

The magnitude of the component of vector a that acts perpendicular to vector b, in the plane formed by the two vectors, is given by

∣a∣sinθ=∣b∣∣a×b∣

where θ is the angle between the two vectors.

Properties of the vector product

AHL 3.16

For any vectors u, v, and w, and scalar k:

u×v=−(v×u)🚫

u×(v+w)=u×v+u×w🚫

(ku)×v=k(u×v)=u×(kv)🚫

u×u=0🚫

u×v=0⇒u∥v for non-zerou,v🚫

Scalar product

AHL 3.13

v⋅w=v1w1+v2w2+v3w3📖

This operation combines corresponding components of each vector, resulting in a single numerical value.

Angle between vectors

AHL 3.13

The scalar product also has a geometric interpretation involving the angle θ between two vectors:

v⋅w=∣v∣∣w∣cosθ📖

Equivalently, isolating cosθ:

cosθ=∣v∣∣w∣v1w1+v2w2+v3w3📖

The angle θ is measured between the heads of v and w:

Acute vs Obtuse Angles

AHL 3.13

If the scalar product of two vectors is negative, then

cosθ=∣u∣⋅∣v∣u⋅v<0

and thus θ must be an obtuse angle: 90°<θ≤180°.

But since vectors always form an acute AND an obtuse angle, we can find the acute angle by subtracting

180°−θ

whenever θ>90°.

Perpendicular vectors

AHL 3.13

Since v⋅w=∣v∣∣w∣cosθ, if two vectors are perpendicular then cos90°=0⇒ their scalar product is 0.

Properties of the scalar product

AHL 3.13

For any vectors u, v, and w, and scalar k:

u⋅v=v⋅u🚫

u⋅(v+w)=u⋅v+u⋅w🚫

(ku)⋅v=k(u⋅v)=u⋅(kv)🚫

u⋅u=∣u∣2🚫

Vector product

AHL 3.16

The vector product, sometimes called the cross product, of two vectors v and w is given by:

v×w=⎝⎛v2w3−v3w2v3w1−v1w3v1w2−v2w1⎠⎞📖

Vector areas with cross product

AHL 3.16

The magnitude of the vector product v×w gives the area of the parallelogram formed by vectors v and w:

Area=∣v×w∣📖

Intuitively, this happens because the magnitude combines both vectors’ lengths and how "spread out" they are from each other, capturing exactly the amount of two-dimensional space they span.

Vector product and sin of angle

AHL 3.16

The magnitude of the vector product is connected to the sine of the angle between the vectors by the formula

∣v×w∣=∣v∣∣w∣sinθ📖

Here, θ is the angle between vectors v and w. This relationship holds because the area of the parallelogram formed by the two vectors depends on their lengths and the angle separating them.

Specifically, the area is largest when the vectors are perpendicular (sin90∘=1) and zero when they are parallel (sin0∘=0).

Components of vectors

AHL 3.16

For two vectors a and b, the magnitude of the component of vector a that acts in the direction of vector b is given by

∣a∣cosθ=∣b∣a⋅b

The magnitude of the component of vector a that acts perpendicular to vector b, in the plane formed by the two vectors, is given by

∣a∣sinθ=∣b∣∣a×b∣

where θ is the angle between the two vectors.

Properties of the vector product

AHL 3.16

For any vectors u, v, and w, and scalar k:

u×v=−(v×u)🚫

u×(v+w)=u×v+u×w🚫

(ku)×v=k(u×v)=u×(kv)🚫

u×u=0🚫

u×v=0⇒u∥v for non-zerou,v🚫