Vector vecv1 points along the z axis and has magnitude v1 78...

Vector $\vec{V_1}$ points along the z axis and has magnitude $V_1 = 78$. Vector $\vec{V_2}$ lies in the xz plane, has magnitude $V_2 = 56$, and makes a -50° angle with the x axis (points below x axis). What is the scalar product $\vec{V_1} \cdot \vec{V_2}$? Express your answer using two significant figures. $\vec{V_1} \cdot \vec{V_2} =$

Transcript

00:02 So in this question we are given information about two vectors.

00:06 V1 is defined along z axis and has a magnitude of 75.

00:12 And vector v2 is in the x5 plane and has a magnitude of 58 and has a direction minus 48 degrees, that is 48 degrees below x axis.

00:24 And we are asked what is the value of v1 dot product with v2.

00:30 What is it? so there is a short answer to this and there's a long answer to this and we'll do both here.

00:36 The short answer is that it is zero because these two vectors are perpendicular to each other.

00:44 And if we know the property that v1 .v2 is zero when v1 is perpendicular to v2, that's by definition because this v1 .2 if we take the definition is equal to, v1 magnitude v2 magnitude into cost of theta between them and if theta is 90 cost theta is 0.

01:13 So this is the short answer and that is because the v1 is along x -axis and v2 is in the x -y -plane.

01:19 So whatever direction v2 is in in the x -y -plane, it is always going to be perpendicular to the x -axis by definition of x -y -z coordinates.

01:27 And so the short answer is that the dot product will be zero.

01:30 But let's work it out because we can't leave the question so easily.

01:34 So let's figure out how to draw these two vectors or get these two vectors.

01:40 So v1 vector is simple, it's 75 k hat and that's because it is along z axis, so that's simply one component is there.

01:52 But the v2 vector is a little complicated, let's just find out how to get that.

01:57 Let's say this is the xy plane and we know that it is 48 degrees below x -axis and its magnitude is 58.

02:09 This is v2.

02:10 So we can find out the components of v1 and v2.

02:13 So this would be the x component...

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