Let $p = \hat{i} -\hat{j} + 2\hat{k}$ and $q = 2\hat{i} +2\hat{j} + 3\hat{k}$.Which of the following statements is correct?

Question

BlackTom AI · Accepted Answer

We start by restating the given vectors p and q to keep the setup clear: p = i_hat - j_hat + 2 k_hat and q = 2 i_hat + 2 j_hat + 3 k_hat.

To determine whether the angle between p and q is acute (less than pi/2), we examine their dot product. The dot product p · q equals (1)(2) + (-1)(2) + (2)(3) = 2 - 2 + 6 = 6, which is a positive number.

A positive dot product indicates that the cosine of the angle between the vectors is positive, since cos(theta) = (p · q) / (|p| |q|). Therefore, theta must lie in the interval (0, pi/2), i.e., the angle is acute.

For completeness, compute the norms: |p| = sqrt(1^2 + (-1)^2 + 2^2) = sqrt(1 + 1 + 4) = sqrt(6) and |q| = sqrt(2^2 + 2^2 + 3^2) = sqrt(4 + 4 + 9) = sqrt(17). The denominator |p| |q| = sqrt(6) * sqrt(17) = sqrt(102).

Hence cos(theta) = 6 / sqrt(102), which is a positive value, confirming theta is between 0 and pi/2. Numerically, 6 / sqrt(102) ≈ 0.594, corresponding to theta ≈ 0.93 radians ≈ 53.5 degrees, well below pi/2.

Given the provided option states that 'The angle between p and q is less than pi/2,' this aligns with the calculation above.

If there were other answer choices, you would apply the same dot-product criterion: a positive dot product implies an acute angle; a zero dot product implies a right angle; a negative dot product implies an obtuse angle. Each option would be evaluated accordingly by computing p · q and comparing signs, or by comparing squared norms if needed for indirect checks.

Let \(p = \hat{i} -\hat{j} + 2\hat{k}\) and \(q = 2\hat{i} +2\hat{j} + 3\hat{k}\).Which of the following statements is correct?单项选择题

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