Vectors & Vector Operations II Quiz, Explanations and Step-by-Step Review

Cross product: perpendicular direction and area

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.

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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

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 \(=0\)).} \]

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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

Key idea

The projection of \(a\) onto a nonzero vector \(b\) is \[ \mathrm{proj}_b a = \frac{a\cdot b}{b\cdot b}\,b \quad (b≠ 0). \] 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: \[ \mathrm{comp}_b a = \frac{a\cdot b}{\|b\|} \quad (b≠ 0). \] A useful decomposition is \[ a = \mathrm{proj}_b a + \bigl(a - \mathrm{proj}_b a\bigr), \] where the second term is perpendicular to \(b\).

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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

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 \(d=0\).

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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

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: \[ e_j=\frac{u_j}{\|u_j\|}\quad (u_j≠ 0). \] The “magic step” is the orthogonal component: \[ v_j - \mathrm{proj}_{u_1}v_j \] (which removes the part of \(v_j\) pointing along \(u_1\)).

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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

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 \[ u\times v=0 \quad \Longleftrightarrow \quad u \text{ and } v \text{ are linearly dependent (parallel) or one is } 0. \]

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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

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

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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.