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Chapter 3: Problem 19

(a) What's the magnitude of \(\hat{\imath}+\hat{\jmath} ?\) (b) What angle does it make with the \(x\) -axis?

Short Answer

Expert verified
The magnitude of the vector \(\hat{\imath}+\hat{\jmath}\) is \(\sqrt{2}\) and the angle it makes with the x-axis is 45 degrees.

Step by step solution

01

Calculate the magnitude (part a)

The given vector is \(\hat{\imath}+\hat{\jmath}\). Here \(\hat{\imath}\) and \(\hat{\jmath}\) are unit vectors along the x and y directions. The magnitude of the resultant vector is given by \(\sqrt{\|\hat{\imath}\|^2 + \|\hat{\jmath}\|^2}\) = \(\sqrt{1^2 + 1^2} = \sqrt{2}\).
02

Calculate the angle (part b)

We can use trigonometric relations to find the angle. The tangent of the angle that the vector makes with the x-axis is the ratio of the y-coordinate of the vector to the x-coordinate. For the given vector, the y-coordinate and the x-coordinate are both 1, so \(\tan(\theta)\) = \(\frac{1}{1} = 1\). To find the angle, take the arctangent (or inverse tangent) on both sides. So \(\theta = \arctan(1)\) = 45 degrees.

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