Mathematics

Mathematics

Problem

Let a⃗=2i+pj+3k\vec a = 2\mathbf{i} + p\mathbf{j} + 3\mathbf{k} and b⃗=i−j+k\vec b = \mathbf{i} - \mathbf{j} + \mathbf{k}. It is known that the length of the projection of a⃗\vec a on b⃗\vec b equals 43\dfrac{4}{\sqrt{3}} units. a. (i) Find the possible value(s) of the real parameter pp. b. (ii) For each value of pp obtained in part (i), determine the angle (to the nearest degree) between a⃗\vec a and b⃗\vec b. c. (iii) State, with reasons, whether each angle found in part (ii) is acute or obtuse.
Intuitive explanation

The projection of one vector onto another measures how much of the first vector points in the direction of the second. You'll use the formula: projection length = a⃗⋅b⃗∣b⃗∣\frac{\vec a \cdot \vec b}{|\vec b|}. Calculate the dot product a⃗⋅b⃗\vec a \cdot \vec b (which will contain the unknown pp), find ∣b⃗∣|\vec b|, set up the equation equal to 43\frac{4}{\sqrt{3}}, and solve for pp.

Intuitive explanation

The projection of one vector onto another measures how much of the first vector points in the direction of the second. You'll use the formula: projection length = a⃗⋅b⃗∣b⃗∣\frac{\vec a \cdot \vec b}{|\vec b|}. Calculate the dot product a⃗⋅b⃗\vec a \cdot \vec b (which will contain the unknown pp), find ∣b⃗∣|\vec b|, set up the equation equal to 43\frac{4}{\sqrt{3}}, and solve for pp.

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