Difficulty: Foundational | Prerequisites: Basic vector algebra from Calculus II
Vectors in three-dimensional space are the building blocks for nearly every problem on Midterm 1. The dot product tells you about angles and projections, the cross product gives you perpendicular vectors and areas, and the scalar triple product extends these ideas to volumes. If you can compute these three operations fluently and interpret what they mean geometrically, you have the foundation for planes, curves, and surfaces.
Vector addition
Given vectors u and v in R³, their sum u + v is obtained by adding corresponding components. In simple terms, you stack the arrows tip-to-tail and draw the shortcut from start to finish.
Dot product (u · v)
The sum of the products of corresponding components: u₁v₁ + u₂v₂ + u₃v₃. It returns a scalar (a single number), not a vector. Think of it as a measure of how much two vectors point in the same direction.
Cross product (u × v)
A vector operation that returns a new vector perpendicular to both u and v, with magnitude equal to the area of the parallelogram they span. In simple terms, the cross product finds the direction that sticks straight out of the plane formed by the two input vectors.
Scalar triple product
The quantity v₁ · (v₂ × v₃), which equals the signed volume of the parallelepiped spanned by the three vectors. Think of it as the 3D analogue of the cross product's area: it tells you how much volume three vectors enclose.
Orthogonality
Two vectors are orthogonal (perpendicular) when their dot product equals zero. In simple terms, they meet at a right angle and have no component in each other's direction.
Parallelepiped
The three-dimensional figure formed by three vectors sharing a common origin, like a skewed box. Its volume is given by the absolute value of the scalar triple product.
The dot product u · v = u₁v₁ + u₂v₂ + u₃v₃ is a scalar.
It relates to the angle θ between the vectors: u · v = |u| |v| cos θ.
If u · v = 0, the vectors are perpendicular.
If u · v > 0, the angle between them is acute (less than 90°).
If u · v < 0, the angle is obtuse (greater than 90°).
The projection of u onto v uses the dot product: proj_v u = ((u · v) / |v|²) v.
The cross product u × v = ⟨u₂v₃ − u₃v₂, u₃v₁ − u₁v₃, u₁v₂ − u₂v₁⟩.
The result is a vector perpendicular to both u and v.
Its magnitude |u × v| equals the area of the parallelogram spanned by u and v.
The direction follows the right-hand rule: curl your fingers from u toward v, and your thumb points in the direction of u × v.
Order matters: u × v = −(v × u). Swapping the order flips the direction.
The vector sum u + v + w can represent the sides of a triangle when the vectors form a closed loop, which is useful for revealing symmetries.
The area of a triangle with sides u and v is ½|u × v|.
The cross product is the go-to tool for computing areas of parallelograms and triangles in 3D.
Volume = |v₁ · (v₂ × v₃)|
For v₁ = ⟨a, b, c⟩, v₂ = ⟨d, e, f⟩, v₃ = ⟨g, h, i⟩:
Volume = |a(ei − fh) + b(gf − di) + c(dh − eg)|
This is equivalent to the absolute value of the 3×3 determinant formed by the three vectors as rows (or columns).
If the scalar triple product equals zero, the three vectors are coplanar (they all lie in the same plane).
\mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 + u_3 v_3\mathbf{u} \cdot \mathbf{v} = |\mathbf{u}|\,|\mathbf{v}|\cos\theta\mathbf{u} \times \mathbf{v} = \langle u_2 v_3 - u_3 v_2,\; u_3 v_1 - u_1 v_3,\; u_1 v_2 - u_2 v_1 \rangle|\mathbf{u} \times \mathbf{v}| = |\mathbf{u}|\,|\mathbf{v}|\sin\theta\text{Volume of parallelepiped} = |\mathbf{v}_1 \cdot (\mathbf{v}_2 \times \mathbf{v}_3)|\text{Area of triangle} = \tfrac{1}{2}|\mathbf{u} \times \mathbf{v}|Students often confuse the dot product and the cross product. The dot product returns a scalar; the cross product returns a vector. They measure different things (alignment vs. perpendicularity).
Students sometimes forget that the cross product is order-dependent: u × v = −(v × u). Swapping the inputs reverses the direction of the resulting vector.
A common error is treating |u × v| as the area of a triangle. It gives the area of the parallelogram. For the triangle, you need to halve it.
Students occasionally assume that if two vectors have the same magnitude, their dot product must be positive. It depends on the angle between them, not their lengths.
⚠️ The dot product formula and its angle interpretation appear on nearly every MATH 215 midterm. Know both forms cold.
⚠️ Cross product computation via the component formula is a frequent exam question. Practise the determinant expansion until it is automatic.
⚠️ The scalar triple product (parallelepiped volume) often appears as a "compute this volume" problem. The key step is computing the cross product first, then dotting the result with the third vector.
⚠️ Expect a question asking you to determine whether vectors are perpendicular, parallel, or neither. Perpendicular: dot product = 0. Parallel: cross product = 0 (zero vector).
True or false: the dot product of two vectors always returns a vector.
False. The dot product returns a scalar.
Fill in the blank: if u · v = 0, the vectors are ______.
Perpendicular (orthogonal).
True or false: u × v = v × u.
False. u × v = −(v × u).
Fill in the blank: the magnitude of the cross product |u × v| gives the area of the ______ spanned by u and v.
Parallelogram.
True or false: if the scalar triple product of three vectors is zero, the vectors are coplanar.
True.
Q: Given u = ⟨1, 2, 3⟩ and v = ⟨4, −1, 2⟩, compute u · v.
A: u · v = (1)(4) + (2)(−1) + (3)(2) = 4 − 2 + 6 = 8.
Q: Using the same vectors, compute u × v.
A: u × v = ⟨(2)(2) − (3)(−1), (3)(4) − (1)(2), (1)(−1) − (2)(4)⟩ = ⟨4 + 3, 12 − 2, −1 − 8⟩ = ⟨7, 10, −9⟩.
Q: Find the area of the parallelogram spanned by u = ⟨1, 0, 3⟩ and v = ⟨2, 1, 0⟩.
A: u × v = ⟨(0)(0) − (3)(1), (3)(2) − (1)(0), (1)(1) − (0)(2)⟩ = ⟨−3, 6, 1⟩. Area = |u × v| = √(9 + 36 + 1) = √46.
Q: Determine whether u = ⟨2, −1, 5⟩ and v = ⟨1, 2, 0⟩ are perpendicular.
A: u · v = (2)(1) + (−1)(2) + (5)(0) = 2 − 2 + 0 = 0. Yes, they are perpendicular.
Q: Find the volume of the parallelepiped determined by v₁ = ⟨1, 0, 2⟩, v₂ = ⟨3, 1, 0⟩, v₃ = ⟨0, 2, 1⟩.
A: First compute v₂ × v₃ = ⟨(1)(1) − (0)(2), (0)(0) − (3)(1), (3)(2) − (1)(0)⟩ = ⟨1, −3, 6⟩. Then v₁ · (v₂ × v₃) = (1)(1) + (0)(−3) + (2)(6) = 1 + 0 + 12 = 13. Volume = |13| = 13.
The dot product reappears when you compute tangent planes (the normal vector to a plane is read off the gradient, and testing perpendicularity uses the dot product). The cross product is essential for finding normal vectors to planes defined by two direction vectors, which connects directly to the Planes and Level Sets topic.
The scalar triple product extends naturally to determinants, which surface again in Jacobians later in the course. Building comfort with these operations now pays off repeatedly.
vector operations, dot product, scalar product, inner product, cross product, vector product, orthogonal vectors, perpendicular vectors, parallelogram area, parallelepiped volume, scalar triple product, determinant, right-hand rule, vector addition, component form, angle between vectors, MATH 215 Midterm 1, Calculus III, multivariable calculus, University of Michigan