Finding the angle between a line and a plane

A shipping container is twelve metres long, three metres wide and three metres tall. A straight cable is stretched from one bottom corner to the top corner diagonally opposite. What angle does the cable make with the floor? Every length you need is inside the box, but the triangle that holds the angle isn't drawn anywhere. Before any trigonometry, you draw one line yourself, straight down onto the floor. That line is what makes the triangle.

This is video ten of eleven in Pythagoras, Trigonometry and Coordinate Geometry. The chains of flat triangles used here are built in Finding lengths in three-D using Pythagoras and trigonometry, so revisit that if they feel shaky.

The angle between a line and a plane is Higher content, and Extended content on Core and Extended courses.

A plane is a flat surface, like the container's floor, stretching out in every direction. The angle between a line and a plane is measured at the point where the line meets it. From the line's top end, drop a perpendicular straight down onto the plane. A perpendicular is a line meeting the plane at ninety degrees, and the point where it lands is called its foot. Now, imagine the sun directly overhead. The cable's shadow on the floor runs from where the cable meets the floor to the foot of that drop, and that shadow is called the projection. Is the shadow longer or shorter than the cable itself? Shorter, unless the cable lies flat. The cable is the longest side of the triangle it makes with its shadow and the perpendicular. So, the angle between a line and a plane is the angle between the line and its shadow, at the point where they meet. Why the shadow? It's the one line on the floor lying directly under the cable, so it measures how steeply the cable really rises. Any other floor line through that point swings away, and gives a bigger angle. That gives three stages, every time. Draw the line. Drop the perpendicular from its top end. Join the foot to where the line meets the plane, and lift out the triangle. Where does the angle sit: the line's top, the foot of the drop, or where the line meets the plane? Where the line meets the plane, between the line and its shadow. The foot of the drop holds the right angle instead.

In the container, the cable meets the floor at a bottom corner, and its top end is the top corner diagonally opposite. Drop straight down from that top corner. The perpendicular is the container's upright edge, three metres, and its foot is the far bottom corner. Join that foot back to the starting corner. That's the floor diagonal - the cable's shadow. Lift out the triangle: the floor diagonal along the bottom, the three-metre edge standing up with a square corner at its foot, and the cable as the longest side. The angle, theta, is at the starting corner. Every angle faces its own side - look straight across. Stand in theta, where the cable meets the floor, and look across: you see the line you dropped. The perpendicular is always the side opposite the angle. The floor diagonal isn't given, so Pythagoras on the floor comes first. Twelve squared is a hundred and forty-four, three squared is nine, and together they make a hundred and fifty-three. The floor diagonal is root a hundred and fifty-three metres, kept as a root. Opposite three, adjacent root a hundred and fifty-three. Which ratio finds theta? The tan ratio, because it pairs the opposite with the adjacent. Tan theta equals three over root a hundred and fifty-three. To find the angle, use the inverse. Press shift, tan, three, divide, root a hundred and fifty-three, close the bracket, equals. The display reads thirteen point six three three zero two two two three. To one decimal place, the cable makes an angle of thirteen point six degrees with the floor. A student skips the drop and uses the container's twelve-metre length as the bottom of the triangle, writing tan theta equals three over twelve. What did that student get wrong about the bottom side? That edge doesn't lie under the cable. The cable heads to the diagonally opposite corner, so what lies directly beneath it is the floor diagonal.

A square-based pyramid has a base of side six metres. Its apex, the top point, is eight metres straight above the centre of the base. The line, this time, is a sloping edge, from a base corner up to the apex. The plane is the base. Which line on the base is the sloping edge's shadow? It runs from where the edge meets the base to the centre, directly under the top point. That's half the base diagonal. The drop from the apex lands on that centre, so the perpendicular is the pyramid's eight-metre height. Lift out the triangle: half the diagonal along the base, the eight-metre height standing up from the centre with the square corner at its foot, and the sloping edge as the longest side. Theta sits at the base corner. Half the diagonal, though, needs Pythagoras first. From the centre, a corner is three metres across and three metres along, so three squared plus three squared is nine plus nine, eighteen. Half the diagonal is root eighteen metres, kept as a root for the next step. Now find theta yourself: opposite eight, adjacent root eighteen. What angle do you get? Tan theta equals eight over root eighteen. Press shift, tan, eight, divide, root eighteen, close the bracket, equals, and the display reads sixty-two point zero six one six four seven two seven. To one decimal place, the sloping edge makes an angle of sixty-two point one degrees with the base. The angle at the apex, between the edge and the height, is the triangle's other angle. It's ninety take away sixty-two point one, twenty-seven point nine degrees, and it answers a different question.

Right, the cable in the container needed one line drawn before any trigonometry. That step now goes onto solids you haven't met. To find the angle between a line and a plane, what do you draw first? The perpendicular from the top of the line straight down onto the plane, then the join from its foot back to where the line meets the plane. Now a cube of side two metres. What angle does its space diagonal make with the base? About thirty-five point three degrees. The floor diagonal is root eight, and tan theta is two over root eight. One more: a box eight by six, five metres tall. Angle of its space diagonal with the floor? Twenty-six point six degrees. Eight, six and ten make a right-angled triangle on the floor, so the angle's tan is five over ten, a half.

When you feel you've fully understood it, a thumbs-up marks this video as done, and it drops out of your revision round. If it hasn't settled, keep it saved for later, then sketch that drop line on three different boxes. After three, drawing it becomes the first thing your pencil does.

Next in the chapter: Trigonometry in three-D and complex figures.

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