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Vectors & Vector Operations II Practice Quiz with a Step-by-Step Interactive Lesson
Use the quiz at the top of the page to practice vectors and vector operations at the next level: the cross product in \(\mathbb{R}^3\) (including right-hand rule direction), area of a parallelogram and area of a triangle via \(\|u\times v\|\), the scalar triple product (also called the mixed product) for volume of a parallelepiped, coplanar vectors and the condition \((u\times v)\cdot w=0\), vector projection and scalar projection (components along a direction), distance from a point to a line/axis and distance from a point to a plane, and the Gram–Schmidt process to build an orthonormal basis. If you want a refresher with worked examples, click Start lesson.
How this vectors practice works
- 1. Take the quiz: answer the vectors and vector operations II questions at the top of the page.
- 2. Open the lesson (optional): review cross product and triple product geometry, projection and scalar component, distance formulas, and Gram–Schmidt orthogonalization.
- 3. Retry: return to the quiz and apply the correct vector formulas immediately.
What you’ll learn in the vectors & vector operations II lesson
Cross product & area in \(\mathbb{R}^3\)
- Cross product computation: \(u\times v\) component formula and determinant form
- Perpendicular vectors and the right-hand rule direction
- Area: \(\|u\times v\|\) (parallelogram) and \(\dfrac12\|u\times v\|\) (triangle)
Scalar triple product, determinants & volume
- Scalar triple product: \((u\times v)\cdot w=\det[u\;v\;w]\)
- Volume of a parallelepiped: \(\left|(u\times v)\cdot w\right|\)
- Coplanarity test: \((u\times v)\cdot w=0\) (volume \(=0\))
Projection, scalar component & distances
- Vector projection: \(\mathrm{proj}_b a=\dfrac{a\cdot b}{b\cdot b}\,b\) and scalar projection: \(\mathrm{comp}_b a=\dfrac{a\cdot b}{\|b\|}\)
- Distance to a line/axis: \(\|a-\mathrm{proj}_d a\|\) (or \(\dfrac{\|a\times d\|}{\|d\|}\))
- Distance to a plane using a normal vector: \(\dfrac{|n\cdot a-d|}{\|n\|}\)
Gram–Schmidt & orthonormal bases
- Gram–Schmidt process: build an orthogonal then orthonormal set
- Orthogonal component: subtract projections step-by-step
- Why it matters: clean coordinates, stable geometry, and foundations for QR decomposition
Back to the quiz
When you’re ready, return to the quiz at the top of the page and keep practicing vectors and vector operations II.
& Vector Ops II
Lesson overview
Purpose: Master Vectors & Vector Operations II in \(\mathbb{R}^3\): compute the cross product for perpendicular directions and area, use the scalar triple product (mixed product) for volume and coplanarity, apply projection and scalar projection, solve distance problems (point-to-line/axis/plane), and run the Gram–Schmidt process to build an orthonormal basis.
Success criteria
- Compute the cross product \(u\times v\) and interpret it as a vector perpendicular to \(u\) and \(v\).
- Use \(\|u\times v\|\) for area of a parallelogram and \(\dfrac12\|u\times v\|\) for area of a triangle.
- Compute the scalar triple product \((u\times v)\cdot w\) and interpret \(\left|(u\times v)\cdot w\right|\) as volume of a parallelepiped.
- Use \((u\times v)\cdot w=0\) to test whether \(u,v,w\) are coplanar (volume \(=0\)).
- Compute vector projection and scalar projection to find components along a direction.
- Compute distance from a point to a line/axis and distance from a point to a plane using dot/cross products.
- Use the Gram–Schmidt process to turn independent vectors into an orthonormal set.
- Recognize quick tests for parallel and perpendicular vectors in \(\mathbb{R}^3\).
Key vocabulary
- Cross product: \(u\times v\) (in \(\mathbb{R}^3\)) gives a vector perpendicular to both \(u\) and \(v\).
- Parallelogram area: \(\|u\times v\|\).
- Scalar triple product: \((u\times v)\cdot w=\det[u\;v\;w]\).
- Parallelepiped volume: \(\left|(u\times v)\cdot w\right|\).
- Coplanar: vectors lie in the same plane; in 3D this occurs when the triple product is \(0\).
- Projection: the “shadow” of one vector onto another direction.
- Gram–Schmidt: algorithm that subtracts projections to make vectors orthogonal, then normalizes to unit length.
Quick pre-check
Cross product: perpendicular direction and area
Learning goal: Compute \(u\times v\) and use \(\|u\times v\|\) to get the area of a parallelogram (and half for a triangle).
Key idea
For \(u=(u_1,u_2,u_3)\) and \(v=(v_1,v_2,v_3)\) in \(\mathbb{R}^3\), the cross product is \[ u\times v= \bigl(u_2v_3-u_3v_2,\;u_3v_1-u_1v_3,\;u_1v_2-u_2v_1\bigr). \] The vector \(u\times v\) is perpendicular to both \(u\) and \(v\). Its magnitude satisfies \[ \|u\times v\|=\|u\|\,\|v\|\sin\theta, \] so \(\|u\times v\|\) is the area of the parallelogram spanned by \(u\) and \(v\), and \(\dfrac12\|u\times v\|\) is the area of the triangle.
Worked example
Example: Find a vector perpendicular to both \((1,0,1)\) and \((0,1,1)\).
Compute the cross product: \[ (1,0,1)\times(0,1,1)= \bigl(0\cdot 1-1\cdot 1,\;1\cdot 0-1\cdot 1,\;1\cdot 1-0\cdot 0\bigr)=(-1,-1,1). \] So \((-1,-1,1)\) is perpendicular to both vectors (and any nonzero scalar multiple also works).
Try it
Summary
- \(u\times v\) is perpendicular to both \(u\) and \(v\) (in \(\mathbb{R}^3\)).
- \(\|u\times v\|\) gives parallelogram area; half of it gives triangle area.
Scalar triple product: volume and coplanarity
Learning goal: Use \((u\times v)\cdot w\) to compute volume and test coplanarity.
Key idea
The scalar triple product (mixed product) is \[ (u\times v)\cdot w. \] It equals the determinant \[ (u\times v)\cdot w = \det[u\;v\;w], \] and its absolute value is the volume of the parallelepiped spanned by \(u,v,w\): \[ \text{Volume}=\left|(u\times v)\cdot w\right|. \] In particular, \[ (u\times v)\cdot w=0 \quad \Longleftrightarrow \quad u,v,w\ \text{are coplanar (volume \(=0\)).} \]
Worked example
Example: What is the volume of the parallelepiped spanned by \((1,2,0)\), \((0,1,2)\), and \((2,0,1)\)?
Let \(a=(1,2,0)\), \(b=(0,1,2)\), \(c=(2,0,1)\). Compute the determinant with these as columns: \[ \det\begin{pmatrix} 1 & 0 & 2\\ 2 & 1 & 0\\ 0 & 2 & 1 \end{pmatrix} = 1\cdot(1\cdot 1-0\cdot 2)\;-\;0(\cdots)\;+\;2\cdot(2\cdot 2-1\cdot 0) =1+8=9. \] So the volume is \(\left|9\right|=9\).
Try it
Summary
- Scalar triple product: \((u\times v)\cdot w=\det[u\;v\;w]\).
- Volume: \(\left|(u\times v)\cdot w\right|\). Coplanar \(\Leftrightarrow\) triple product \(=0\).
Projection and scalar component along a direction
Learning goal: Project vectors in \(\mathbb{R}^3\) and compute the scalar component (signed length) along a direction.
Key idea
The projection of \(a\) onto a nonzero vector \(b\) is \[ \mathrm{proj}_b a = \frac{a\cdot b}{b\cdot b}\,b \quad (b\neq 0). \] This returns the vector component of \(a\) that points in the direction of \(b\). The scalar projection (also called the scalar component) is the signed length: \[ \mathrm{comp}_b a = \frac{a\cdot b}{\|b\|} \quad (b\neq 0). \] A useful decomposition is \[ a = \mathrm{proj}_b a + \bigl(a - \mathrm{proj}_b a\bigr), \] where the second term is perpendicular to \(b\).
Worked example
Example: What is the projection of \((1,2,3)\) onto \((3,0,0)\)?
Let \(a=(1,2,3)\), \(b=(3,0,0)\). \[ a\cdot b = 1\cdot 3 + 2\cdot 0 + 3\cdot 0 = 3,\quad b\cdot b = 3^2 = 9. \] \[ \mathrm{proj}_b a = \frac{3}{9}(3,0,0)=\left(1,0,0\right). \]
Try it
Summary
- \(\mathrm{proj}_b a=\dfrac{a\cdot b}{b\cdot b}b\) and \(\mathrm{comp}_b a=\dfrac{a\cdot b}{\|b\|}\) (for b≠ 0).
- Projection is the “along \(b\)” part; \(a-\mathrm{proj}_b a\) is perpendicular to \(b\).
Distance to a line/axis and distance to a plane
Learning goal: Turn geometric distance questions into dot/cross product computations.
Key idea
Distance from a point to a line through the origin: if the line is in direction d≠ 0 and the point position vector is \(p\), then the closest point on the line is \(\mathrm{proj}_d p\), so \[ \text{dist}(p,\text{line})=\left\|p-\mathrm{proj}_d p\right\|. \] An equivalent cross-product formula is \[ \text{dist}(p,\text{line})=\frac{\|p\times d\|}{\|d\|}. \] Distance from a point to a plane: for a plane \(n\cdot x=d\) with normal vector n≠ 0, \[ \text{dist}(p,\text{plane})=\frac{|n\cdot p-d|}{\|n\|}. \] If the plane passes through the origin, then \(d=0\).
Worked example
Example: What is the distance from \((1,2,3)\) to the plane through the origin with normal \((1,1,1)\)?
The plane is \(n\cdot x=0\) with \(n=(1,1,1)\). \[ n\cdot p = (1,1,1)\cdot(1,2,3)=1+2+3=6,\quad \|n\|=\sqrt{1^2+1^2+1^2}=\sqrt{3}. \] So \[ \text{dist}=\frac{|6|}{\sqrt{3}}=\frac{6}{\sqrt{3}}=2\sqrt{3}. \]
Try it
Summary
- Point-to-line distance (line through origin): \(\|p-\mathrm{proj}_d p\|\) or \(\dfrac{\|p\times d\|}{\|d\|}\).
- Point-to-plane distance (plane \(n\cdot x=d\)): \(\dfrac{|n\cdot p-d|}{\|n\|}\).
Gram–Schmidt: build an orthonormal basis
Learning goal: Use projections to turn independent vectors into orthogonal (and then orthonormal) vectors.
Key idea
Given linearly independent vectors \(v_1,\dots,v_k\), Gram–Schmidt creates an orthogonal set \(u_1,\dots,u_k\) by subtracting projections: \[ u_1=v_1,\quad u_j = v_j - \sum_{i=1}^{j-1}\mathrm{proj}_{u_i} v_j \quad (j\ge 2). \] Then normalize to get an orthonormal set: \[ e_j=\frac{u_j}{\|u_j\|}\quad (u_j\neq 0). \] The “magic step” is the orthogonal component: \[ v_j - \mathrm{proj}_{u_1}v_j \] (which removes the part of \(v_j\) pointing along \(u_1\)).
Worked example
Example: Run Gram–Schmidt for \(v_1=(1,0,0)\) and \(v_2=(1,1,0)\).
Start with \(u_1=v_1=(1,0,0)\). Compute the projection of \(v_2\) onto \(u_1\): \[ \mathrm{proj}_{u_1}v_2=\frac{v_2\cdot u_1}{u_1\cdot u_1}\,u_1 =\frac{(1,1,0)\cdot(1,0,0)}{(1,0,0)\cdot(1,0,0)}(1,0,0) =\frac{1}{1}(1,0,0)=(1,0,0). \] So the orthogonal component is \[ u_2=v_2-\mathrm{proj}_{u_1}v_2=(1,1,0)-(1,0,0)=(0,1,0). \] These are already unit vectors, so \(e_1=(1,0,0)\), \(e_2=(0,1,0)\).
Try it
Summary
- Subtract projections to remove “overlap” and create orthogonal vectors.
- Normalize to get an orthonormal basis: divide by the magnitude.
Angles, parallel vectors, and quick tests
Learning goal: Compute the angle between vectors and recognize parallelism in \(\mathbb{R}^3\).
Key idea
The angle \(\theta\) between nonzero vectors \(u\) and \(v\) satisfies \[ \cos\theta=\frac{u\cdot v}{\|u\|\,\|v\|}. \] Perpendicular means \(u\cdot v=0\). Parallel (same or opposite direction) means one is a scalar multiple of the other. In \(\mathbb{R}^3\), a powerful parallel test is \[ u\times v=0 \quad \Longleftrightarrow \quad u \text{ and } v \text{ are linearly dependent (parallel) or one is } 0. \]
Worked example
Example: What is the value of \(\cos\theta\) between \((1,2,2)\) and \((2,2,1)\)?
Compute the dot product: \[ (1,2,2)\cdot(2,2,1)=1\cdot 2+2\cdot 2+2\cdot 1=2+4+2=8. \] Compute magnitudes: \[ \|(1,2,2)\|=\sqrt{1^2+2^2+2^2}=\sqrt{9}=3,\quad \|(2,2,1)\|=\sqrt{2^2+2^2+1^2}=\sqrt{9}=3. \] So \[ \cos\theta=\frac{8}{3\cdot 3}=\frac{8}{9}. \]
Try it
Summary
- Angle formula: \(\cos\theta=\dfrac{u\cdot v}{\|u\|\|v\|}\) (for nonzero vectors).
- Parallel in \(\mathbb{R}^3\): \(u\times v=0\) (and nonzero vectors are scalar multiples).
Why these vector operations matter
Learning goal: Connect advanced vector operations to geometry and applications — then finish with a final check.
Where vectors & vector operations II show up
- 3D geometry: areas, volumes, coplanarity, and distances to lines/planes.
- Physics and engineering: torque (\(r\times F\)), work and components (projections), and normal directions.
- Computer graphics: surface normals via cross products, lighting via dot products, and camera geometry.
- Linear algebra: orthonormal bases and the Gram–Schmidt process (a foundation for QR and least squares).
Worked example: scalar component in a direction
Example: What is the scalar component of \((3,4,0)\) in the direction of \((0,1,0)\)?
The direction vector is \(b=(0,1,0)\), which already has \(\|b\|=1\). \[ \mathrm{comp}_b a=\frac{a\cdot b}{\|b\|}=\frac{(3,4,0)\cdot(0,1,0)}{1}=4. \]
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Final recap
- Cross product: \(u\times v\) is perpendicular to \(u\) and \(v\); \(\|u\times v\|\) is parallelogram area.
- Scalar triple product: \((u\times v)\cdot w=\det[u\;v\;w]\); \(\left|(u\times v)\cdot w\right|\) is volume; \(=0\) means coplanar.
- Projection: \(\mathrm{proj}_b a=\dfrac{a\cdot b}{b\cdot b}b\), scalar projection: \(\mathrm{comp}_b a=\dfrac{a\cdot b}{\|b\|}\).
- Distance: to plane \(n\cdot x=d\) is \(\dfrac{|n\cdot p-d|}{\|n\|}\); to a line through the origin direction \(d\) is \(\|p-\mathrm{proj}_d p\|\).
- Gram–Schmidt: subtract projections to get orthogonal vectors, then normalize for an orthonormal basis.
Next step: Close this lesson and try your quiz again. If you miss a question, reopen the book and review the page that matches the vector skill you need.