Question
Vectors a and b are given by a = 3i − 2j + k and b = −2i + 2j + 4k . Find
i) magniture of a and b and the angle between them.
ii) projection of the vector (a +1/ 2b) onto a .
iii) which of the following vectors are perpendicular to a ?
c = −i − 4j + 2k, d = −3i + k, e = 2i + 2j − 2k .
Answer :
Word Count : 577
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Skip in 10
Let's solve the given problems step by step. ### i) Magnitude of vectors a and b, and the angle between them Given vectors: \[ \mathbf{a} = 3\mathbf{i} - 2\mathbf{j} + \mathbf{k} \] \[ \mathbf{b} = -2\mathbf{i} + 2\mathbf{j} + 4\mathbf{k} \] The magnitude of a vector \(\mathbf{v} = v_x \mathbf{i} + v_y \mathbf{j} + v_z \mathbf{k}\) is given by: \[ |\mathbf{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2} \] 1. Magnitude of \(\mathbf{a}\): \[ |\mathbf{a}| = \sqrt{(3)^2 + (-2)^2 + (1)^2} = \sqrt{9 + 4 + 1} = \sqrt{14} \] 2. Magnitude of \(\mathbf{b}\): \[ |\mathbf{b}| = \sqrt{(-2)^2 + (2)^2 + (4)^2} = \sqrt{4 + 4 + 16} = \sqrt{24} = 2\sqrt{6} \] To find the angle \(\theta\) between vectors a and b, we use the formula: \[ \cos\theta = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}| |\mathbf{b}|} \] where \(\mathbf{a} \cdot \mathbf{b}\) is the dot product of vectors a and b: \[ \mathbf{a} \cdot
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