Applications of Vector Algebra

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Step 1: Use the determinant formula: [a, b, c] = |1 2 3| |2 1 -1| |3 2 1| Step 2: Expand along first ro…

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Step 1: Volume = |[a, b, c]| = |a · (b × c)| Step 2: Calculate the determinant: |2 -3 4| |1 2 -1| |3 -1 2| Step 3: Expand along first row: = 2(2·2 - (-1)·(-1)) - (-3)(1·2 - (-1)·3) + 4(1·(-1…

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Use this condition to check if three vectors are coplanar (lying in the same plane). Context: Three non-zero vectors a, b, c are coplanar if and only if their scalar triple product equals zero. This …

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For vectors a = a₁i + a₂j + a₃k, b = b₁i + b₂j + b₃k, c = c₁i + c₂j + c₃k: [a, b, c] = (a × b) · c = |a₁ a₂ a₃| |b₁ b₂ b₃| |c₁ c₂ c₃| Va…

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Step 1: Use dot product formula: a · b = a₁b₁ + a₂b₂ + a₃b₃ Step 2: Identify components: a = (1, 2, -3), b = (2, -1, 2) Step 3: Calculate: a · b = (1)(2) + (2)(-1) + (-3)(2) = 2 - 2 - 6 …

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Step 1: Use the formula: a × (b × c) = (a · c)b - (a · b)c Step 2: Calculate a · c: a · c = (1)(0) + (-1)(3) + (0)(-1) = 0 - 3 + 0 = -3 Step 3: Calculate a · b: a · b = (1)(1) + (-1)(-1) + (0)(-4) =…

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Step 1: Find the direction vector AB: AB = B - A = (4-1, 6-2, 9-3) = (3, 4, 6) Step 2: Calculate magnitude: |AB| = √(3² + 4² + 6²) = √(9 + 16 + 36) = √61 Step 3: Find direction cosines: l = 3/√61, m…

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Step 1: Identify the given point: (x₁, y₁, z₁) = (2, 3, -1) Step 2: Identify direction ratios: (b₁, b₂, b₃) = (2, 3, -1) Step 3: Use the standard form: (x - x₁)/b₁ = (y - y₁)/b₂ = (z - z₁)/b₃ Step …