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Vectors (Standard)

Multi-step exam-style items.

Phase 2 · JEE

FoundationsStandardAdvanced

Vectors: core idea

a⃗⋅b⃗=∣a⃗∣∣b⃗∣cos⁡θ\vec a\cdot\vec b=|\vec a||\vec b|\cos\theta.

Example problem: hint and answer

⟨1,2,3⟩×⟨0,1,0⟩\langle1,2,3\rangle\times\langle0,1,0\rangle: first component?

Hint: Determinant. Answer: -3.

Practice (9 problems — type your answers)

  • 1. Area of parallelogram spanned by ⟨1,0⟩\langle1,0\rangle, ⟨1,1⟩\langle1,1\rangle?

  • 2. Scalar triple [i,j,k][\mathbf i,\mathbf j,\mathbf k]?

  • 3. Do ⟨1,2⟩\langle1,2\rangle and ⟨−2,1⟩\langle-2,1\rangle perpendicular? 1/01/0

  • 4. Distance from (1,2,3)(1,2,3) to origin?

  • 5. Direction cosines of ⟨1,0,0⟩\langle1,0,0\rangle: sum of squares?

  • 6. Normal to plane x+2y+3z=6x+2y+3z=6: first component of a normal?

  • 7. Midpoint of (0,0,0)(0,0,0) and (2,4,6)(2,4,6): zz?

  • 8. Projection length of ⟨4,0⟩\langle4,0\rangle onto ⟨1,1⟩\langle1,1\rangle?

  • 9. ∣a×b∣|\mathbf a\times\mathbf b| if ∣a∣=∣b∣=1|a|=|b|=1 orthogonal?

Vectors Practice Problems — Standard