Problem Metadata

Date November 17, 2025 (optional — no deadline)
Topic Vector Algebra (Component-wise Operations, Dot Product, Norm)
Sub-Topics Dot product, norm, cosine similarity, scalar projection, vector projection, decomposition, orthogonality, cross product, right-hand rule, geometric area, etc
Curator @Anonymous

Main Problem

Daily Math Problem - DAY 7

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Please check this out first!!

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Difficulty

Let $u = 6i - 2j + 4k$, $v = -3i + 7j -k$

Problem 1 — Angle Between Two Vectors

Compute the angle $\theta$ between $u$ and $v$ using

$$ \cos\theta = \frac{u \cdot v}{\|u\|\, \|v\|} $$

Problem 2 — Projection of One Vector onto Another

Compute the projection of $u$ onto $v$:

$$ \operatorname{proj}_{v}(u) = \frac{u \cdot v}{\|v\|^{2}}\, v. $$

Problem 3 — Component of a Vector Orthogonal to Another

Compute the vector component of $u$ orthogonal to $v$:

$$ u_\perp = u - proj_v(u) $$

Problem 4 — Cross Product (Geometric Interpretation)

Compute the cross product $u \times v$.