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Vectors can't be "multiplied" in the same way scalars can. We've already learned about scalar multiples, but how about multiplying two vectors together to get a third?
Your first instinct might be to take vectors a=⎝⎛a1a2a3⎠⎞ and b=⎝⎛b1b2b3⎠⎞ and multiply them together as
While this does technically constitute a mathematical formula, it's not one that has much use in anything below graduate mathematics since it doesn't tell us much useful information about the original vectors. The more common types of vector multiplication involve specific formulas that multiply vector components to produce scalar or vector quantities that will give us a lot of information about a and b. In this section, we'll explore a few different kinds of vector operations with similarities to multiplication.
The first is the scalar product.
Scalar product
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:
This operation combines corresponding components of each vector, resulting in a single numerical value.
Example
Exercise
Find the scalar product⎝⎛34−8⎠⎞⋅⎝⎛−257⎠⎞.
Select the correct option
The scalar product is quite straightforward to define -- but what does it actually represent? Let's use a simple example to build some understanding.
The diagram below shows two vectors, v and w, as well as their dot product. The vector v is dynamic and can be moved, but its magnitude ∣v∣ always remains fixed at 1, while w is statically defined w=(10), meaning only the angle between them changes. The value of their scalar product will update as you change v.
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Discussion
When does the scalar product achieve its greatest value? Its smallest?
What do you notice about the relationship between the value of v's horizontal component and the scalar product v⋅w?
What does this tell you about how the angle between the vectors impacts the scalar product?
Since ∥v∥ is fixed and w=(1,0) with ∥w∥=1, we have
and because −1≤v1≤1,
The scalar product is
– greatest, 1, when v1=1, i.e. v and w point in the same direction (parallel)
– smallest, −1, when v1=−1, i.e. they point in exactly opposite directions (parallel but opposite)
– zero when v1=0, i.e. the vectors are perpendicular
Thus v⋅w is exactly the horizontal component of v. Since rotating v changes only its angle relative to w, the dot product varies from 1 down to –1 according to that angle. In other words, the scalar product measures how “aligned” the two vectors are: positive for acute, zero for right-angle, negative for obtuse, maximal when they are parallel and minimal when they are opposite.
Angle between vectors
The scalar product also has a geometric interpretation involving the angle θ between two vectors:
Equivalently, isolating cosθ:
The angle θ is measured between the heads of v and w:
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Acute vs Obtuse Angles
If the scalar product of two vectors is negative, then
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
whenever θ>90°.
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Example
Find the acute angle between ⎝⎛123⎠⎞ and ⎝⎛321⎠⎞.
Both vectors have magnitude
and the scalar product is
So
Thus θ=cos−1(75)=44.4°.
Exercise
Given that ∣p∣=3, ∣q∣=5 and the angle between p and q is 120∘, find p⋅q.
Select the correct option
The scalar product basically reflects how "aligned" two vectors are. If they point in the same direction and the angle between them is zero, then their scalar product is the magnitude of the first times the magnitude of the second, which is the greatest possible scalar product. So when they are parallel, they are "maximally aligned"; when they are parallel but opposite, they are "minimally aligned."
When the vectors are perpendicular, their alignments are completely unrelated. In the example diagram above, this meant
since one pointed entirely in the x-direction and the other entirely in the y-direction.
We can extrapolate this idea to any two perpendicular vectors.
Perpendicular vectors
Since v⋅w=∣v∣∣w∣cosθ, if two vectors are perpendicular then cos90°=0⇒ their scalar product is 0.
Example
Find k such that ⎝⎛132⎠⎞ is perpendicular to ⎝⎛−3k13⎠⎞.
Exercise
Find the value(s) of k for which the vector ⎝⎛k+322k−1⎠⎞ is perpendicular to ⎝⎛−322⎠⎞.
Select the correct option
The scalar product has several useful properties that arise directly from its definition. We can discover these easily here by setting up different equations involving the dot product.
Discussion
The vectors v and w are given by v=(v1v2),w=(w1w2). This means their scalar product is
Keeping in mind the definition of a scalar product, can you compute the values of
w⋅v?
kv⋅w, for any k∈R?
v⋅kw, for any k∈R?
v⋅v?
Now consider a third vector u=(u1u2). What are the values of
(u⋅v)+(u⋅w)?
u⋅(v+w)?
w⋅v
since v1w1=w1v1 and v2w2=w2v2, this equals v1w1+v2w2, i.e.\ the same as v⋅w.
(kv)⋅w First kv=(kv1,kv2). Then
v⋅(kw) Similarly kw=(kw1,kw2), so
v⋅v
(u⋅v)+(u⋅w)
u⋅(v+w) First v+w=(v1+w1,v2+w2), so
Notice that the results of the last two are equivalent.
Properties of the scalar product
For any vectors u, v, and w, and scalar k:
Checkpoint
For two vectors u and v, it is given that
Find the value of u⋅v.
Select the correct option
Exercise
Given that ∣u∣=5, ∣v∣=6 and u⋅v=2, find (2u−3v)⋅(21u+v).
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Another multiplication-esque computation involving vectors is the vector product. As the name suggests, the vector product outputs another vector. This equation can seem overly complex and pointless at first, but upon further inspection, its properties are incredibly useful to our study of vectors.
Vector product
The vector product, sometimes called the cross product, of two vectors v and w is given by:
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.
(animation that shows taking the determinant)
Example
Checkpoint
Two vectors are given by v=⎝⎛0−11⎠⎞ and w=⎝⎛10−1⎠⎞.
Find the vector product v×w.
Select the correct option
Exercise
The vectors v and w are defined v=⎝⎛13−2⎠⎞, w=⎝⎛a2b⎠⎞
It is given that v×w=⎝⎛7511⎠⎞.
Find the values of a and b.
Select the correct option
One of the major reasons we place importance on this obtuse formula is because of its relation to the area of the parallelogram formed by the component vectors.
Vector areas with cross product
The magnitude of the vector product v×w gives the area of the parallelogram formed by vectors v and 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.
picture
Exercise
Find the area of the parallelogram with adjacent sides v=5i+j−2k and w=i+2k.
Select the correct option
Since the vector product computes area, it, too, has a relationship to the angle θ between two vectors.
Vector product and sin of angle
The magnitude of the vector product is connected to the sine of the angle between the vectors by the formula
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).
Checkpoint
The vectors v and w have magnitudes ∣v∣=7, ∣w∣=4.
Find the value of the acute angle θ between them for which ∣v×w∣=28.
Select the correct option
This conception of the vector product highlights the main qualities of and differences between scalar and vector products. Both scalar and vector products are ways to "multiply" vectors. There's only one way to multiply scalars together, but since vectors have both a magnitude and a direction, there are multiple ways to "multiply" them.
Scalar Product
The scalar product measures the extent to which two vectors point in the same direction. The bigger the vectors and the more they point in the same direction, the bigger the scalar product, since only the "parallel parts" contribute to the dot product. It's is a scalar quantity measuring the amount of similarity.
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Vector Product
The vector product measures the extent to which two vectors have different directions. The bigger the vectors and the more they point in different directions, the bigger the area between them and therefore the bigger the vector product. Only the "perpendicular parts" contribute to the vector product. It's a vector quantity measuring the direction of difference.
Because the scalar product measures the amount of similarity between two vectors and the vector product measures the direction of difference, we can actually use these formulas to quantify "how much" one vector is acting in the same or opposite direction as another.
Components of vectors
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
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
where θ is the angle between the two vectors.
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We can see easily how much one vector points is in the direction of one of the unit vectors by writing it as a sum of i,j and k, but when we want to know how much one vector points in the direction of a non-unit vector, these formulas are necessary.
Exercise
Two vectors are given by a=⎝⎛10−2⎠⎞,b=⎝⎛21−1⎠⎞.
Find the magnitude of the component of a acting
in the direction of b
perpendicular to b
Select the correct option
The properties of the vector product are somewhat similar to those of the scalar product:
Properties of the vector product
For any vectors u, v, and w, and scalar k:
Discussion
Thinking about the vector product in terms of direction, why does the property u×v=−(v×u) make sense? What does it tell you about what happens when you "flip" the orientation of a shape?
The equation (ku)×v=k(u×v)=u×(kv) tells us that multiplying a vector by k and then taking the cross product is the same as taking k copies of the vector product. What does this tell you about the area of the parallelogram determined by two vectors?
Finally, consider the angle formula ∣v×w∣=∣v∣∣w∣sinθ. How can you use this formula to understand the properties u×u=0 and u×v=0⟹u∣∣v? What does it tell you about the angle θ between vectors?
Part (a)
The key is the right-hand rule for the cross product. If you point your index finger along u and your middle finger along v, your thumb gives the direction of u×v. Swapping the order means your index finger now points along v and your middle along u, so your thumb points the opposite way. Hence
Geometrically, u then v “walks” you around the perimeter of a parallelogram in one sense (say, counter-clockwise), producing a normal vector in one direction. If you flip the order to v then u, you traverse the same parallelogram in the opposite sense (clockwise), so the normal vector reverses. Flipping the orientation of the shape flips the sign of its cross-product.
Part (b)
Imagine a parallelogram with base u and height h. Doubling the base means you can place two copies of the original parallelogram side by side, so the total area doubles. More generally, stretching the base by k “stretches” the shape, fitting ∣k∣ copies of the original parallelogram along that direction, so the new area is ∣k∣ times the original.
Since the area is given by
and (ku)×v=k(u×v), we get
so the area scales by ∣k∣.
This reflects the fact that scaling the length of one vector by k scales ∣u∣ by ∣k∣, and hence scales the cross‐product magnitude (and thus the parallelogram’s area) by ∣k∣.
Part (c)
The vector product’s magnitude measures the “amount of tilting” between two vectors by giving the area of the parallelogram they span; its direction is perpendicular to that parallelogram by the right-hand rule. Intuitively:
If you take the same vector twice ( u and u ), you get a “parallelogram” with no width—it’s just a line. No width = zero area, so u×u=0.
If two nonzero vectors u and v produce u×v=0, that means the parallelogram they would span collapses to a line (zero area). Geometrically the only way for it to collapse is if they lie along the same line—i.e. they point in the same or exactly opposite direction, and therefore are parallel.
Thus “vector product zero” exactly captures “no angular separation” between the vectors. In everyday terms, the vector product measures how much one vector sticks out perpendicular to another; if it sticks out nothing, they must be pointing along the same line.
Checkpoint
For two vectors v and w, it is given that v×(3w)=⎝⎛09−3⎠⎞
Find v×w.
Select the correct option
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