Difficulty: Introductory | Prerequisites: basic vector notation, components in R3
Multivariable calculus introduces three distinct ways to "multiply" vectors, and each one answers a different geometric question. The scalar product scales a vector's length. The dot product measures how much two vectors point in the same direction. The cross product produces a brand-new vector perpendicular to both inputs. You will use all three constantly when working with planes, projections, torque, and surface areas later in the course, so getting comfortable with them now pays off for the rest of MATH 126.
Three operations act on vectors in R3. Scalar multiplication stretches or shrinks a vector without changing its direction. The dot product takes two vectors and returns a single number that encodes the angle between them. The cross product takes two vectors and returns a new vector that is perpendicular to both, with a magnitude equal to the area of the parallelogram the two vectors span.
Scalar product (scalar multiplication)
Multiplying a vector v by a real number c to produce cv =
In simple terms, this means stretching or compressing an arrow without rotating it.
Dot product (inner product)
An operation on two vectors that returns a scalar: v . w = v1w1 + v2w2 + v3w3. Equivalently, v . w = |v||w|cos(theta).
Think of it as a measure of how aligned two vectors are. Parallel vectors give the largest dot product; perpendicular vectors give zero.
Cross product (vector product)
An operation on two vectors that returns a new vector perpendicular to both: v x w is computed via the 3x3 determinant with i, j, k in the top row. Its magnitude is |v||w|sin(theta).
Think of it as the vector that sticks straight out of the plane formed by the two input vectors, sized by the area of the parallelogram they span.
Orthogonal
Two vectors are orthogonal when the angle between them is 90 degrees, i.e. their dot product equals zero.
In simple terms, perpendicular.
Unit vector
A vector of magnitude 1, obtained by dividing any nonzero vector by its own magnitude.
In simple terms, a direction arrow with length exactly 1.
Given a scalar c and a vector v =
Returns a vector
Geometric interpretation: scales the length of the vector by |c| while preserving direction (c > 0) or reversing it (c < 0)
Key uses:
Creating unit vectors: divide any nonzero vector by its magnitude to get a vector of length 1 in the same direction
Detecting parallel vectors: v and w are parallel if and only if one is a scalar multiple of the other
Given vectors v =
Returns a scalar
Geometric interpretation: v . w = |v||w|cos(theta), where theta is the angle between v and w. This equals the magnitude of one vector times the component of the other projected onto it.
Key uses:
Detecting right angles: orthogonal vectors have dot product = 0
Computing the angle between two vectors or lines: theta = arccos(v . w / (|v||w|))
Finding scalar projections and vector projections
Given vectors v and w, the cross product v x w is computed as the determinant of a 3x3 matrix with i, j, k in the first row, v's components in the second, and w's in the third
v x w =
Returns a vector
Geometric interpretation: produces a vector orthogonal to both v and w, with magnitude |v||w|sin(theta)
Key uses:
Generating normal vectors to a plane (given two vectors lying in the plane, their cross product is perpendicular to it)
Computing areas: the area of a parallelogram spanned by v and w equals |v x w|; the area of the triangle is half that
\text{Scalar product: } c\vec{v} = \langle cv_1,\, cv_2,\, cv_3 \rangle\text{Dot product: } \vec{v} \cdot \vec{w} = v_1 w_1 + v_2 w_2 + v_3 w_3 = |\vec{v}|\,|\vec{w}|\cos\theta\text{Cross product: } \vec{v} \times \vec{w} = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ v_1 & v_2 & v_3 \\ w_1 & w_2 & w_3 \end{vmatrix} = \langle v_2 w_3 - v_3 w_2,\; -(v_1 w_3 - v_3 w_1),\; v_1 w_2 - v_2 w_1 \rangle|\vec{v} \times \vec{w}| = |\vec{v}|\,|\vec{w}|\sin\thetaThe dot product is how physicists calculate work done by a force: W = F . d. Only the component of the force in the direction of motion contributes.
The cross product shows up in torque calculations (torque = r x F) and in computing the normal to a surface, which is essential in computer graphics for shading and lighting.
Students often think the dot product returns a vector. It does not. The dot product always returns a scalar (a single number).
Students sometimes confuse the dot product formula with the cross product formula. Remember: dot = sum of component-wise products, cross = determinant.
The cross product is not commutative. v x w = -(w x v). Swapping the order flips the direction of the resulting vector.
Students sometimes believe that if the dot product is negative, the vectors are perpendicular. A negative dot product means the angle between the vectors is greater than 90 degrees. Perpendicularity requires the dot product to be exactly zero.
⚠️ Be ready to compute all three products by hand, especially the cross product determinant. This is a staple exam question.
⚠️ Know the geometric interpretation of each product. Expect questions asking you to find angles between vectors using the dot product.
⚠️ "Find a vector perpendicular to both v and w" is a cross product question.
⚠️ Area of a triangle formed by two vectors = (1/2)|v x w|. This comes up frequently.
True or false: The dot product of two vectors always returns a vector.
False. It returns a scalar.
Fill in the blank: Two vectors are orthogonal if and only if their dot product equals ___.
Zero.
True or false: v x w = w x v.
False. v x w = -(w x v).
Fill in the blank: The magnitude of the cross product |v x w| equals |v||w|sin(___) .
theta (the angle between the vectors).
True or false: Multiplying a vector by a negative scalar reverses its direction.
True.
Q: Given v = <1, 2, 3> and w = <4, -1, 2>, compute v . w.
A: v . w = (1)(4) + (2)(-1) + (3)(2) = 4 - 2 + 6 = 8.
Q: Find a vector perpendicular to both v = <1, 0, 0> and w = <0, 1, 0>.
A: v x w = <(0)(0) - (0)(1), -((1)(0) - (0)(0)), (1)(1) - (0)(0)> = <0, 0, 1>. This is k, the unit vector along the z-axis.
Q: What is the area of the parallelogram formed by v = <3, 0, 0> and w = <0, 4, 0>?
A: v x w = <0, 0, 12>. |v x w| = 12. The area is 12 square units.
Q: Find the angle between v = <1, 1, 0> and w = <1, 0, 0>.
A: cos(theta) = v . w / (|v||w|) = 1 / (sqrt(2) * 1) = 1/sqrt(2). So theta = pi/4, or 45 degrees.
Q: Are v = <2, -3, 1> and w = <1, 2, 4> orthogonal?
A: v . w = (2)(1) + (-3)(2) + (1)(4) = 2 - 6 + 4 = 0. Yes, they are orthogonal.
The dot product reappears when you study projections, work, and directional derivatives later in multivariable calculus. The cross product is essential for finding normal vectors to planes (Chapter 12) and for computing flux integrals and curl in vector calculus (Chapters 16-17). Scalar multiplication underpins parametric equations for lines and curves.
scalar multiplication, dot product, inner product, cross product, vector product, orthogonal, perpendicular, unit vector, angle between vectors, projection, normal vector, parallelogram area, triangle area, determinant, right-hand rule, MATH 126, Calculus III, multivariable calculus, vector operations