Dot Product

Question

Given two vectors a = 2i + 3j - k and b = i - 2j + 4k, what is the value of their scalar product a . b?

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The scalar (dot) product of two vectors expressed in component form is obtained by multiplying corresponding components and adding the results, giving a . b = a_x b_x + a_y b_y + a_z b_z. This operation produces a scalar and geometrically equals |a||b|cos(theta), making it the standard JEE Advanced tool for angle and projection problems. Substituting the components, a . b = (2)(1) + (3)(-2) + (-1)(4) = 2 - 6 - 4 = -8. Option -4 arises from dropping the last term, treating the vectors as two-dimensional. Option 4 comes from taking the absolute values of each product before summing, which is incorrect because sign information must be retained. Option 8 reverses the sign of the final answer through an arithmetic slip. The negative result is significant: it shows the angle between the vectors is obtuse, since cos(theta) shares the sign of the dot product. Plausibility check: |a| = sqrt(14) and |b| = sqrt(21), so |a||b| = sqrt(294) is about 17.1, and -8 lies comfortably within [-17.1, 17.1], confirming the result is geometrically admissible.

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Dot Product

Select the correct option:

🔒 Solution Hidden from View

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Solution

The dot product of vectors $a$ and $b$ is defined as